III a Functional Approach to General Topology

III A Functional Approach to General Topology

Maria Manuel Clementino, Eraldo Giuli and Walter Tholen

In this chapter we wish to present a categorical approach to fundamental con- cepts of General Topology, by providing a category with an additional structure which allows us to display more directly the geometricX properties of the objects of regarded as spaces. Hence, we study topological properties for them, such as HausdorffX separation, compactness and local compactness, and we describe important topological constructions, such as the compact-open topology for function spaces and the Stone-Cechˇ compactification. Of course, in a categorical setting, spaces are not investigated “directly” in terms of their points and neighbourhoods, as in the traditional set-theoretic setting; rather, one exploits the fact that the re- lations of points and parts inside a space become categorically special cases of the relation of the space to other objects in its category. It turns out that many stability properties and constructions are established more economically in the categorical rather than the set-theoretic setting, leave alone the much greater level of generality and applicability. The idea of providing a category with some kind of topological structure is cer- tainly not new. So-called Grothendieck topologies (see Chapter VII) and, more generally, Lawvere-Tierney topologies are fundamental for the geometrically-inspired construction of topoi. Specifically, these structures provide a notion of closure and thereby a notion of closed subobject, for every object in the category, such that all morphisms become “continuous”. The notion of Dikranjan-Giuli closure operator [17] axiomatizes this idea and can be used to study topological properties categorically (see, for example, [9, 12]). Here we go one step further and follow the approach first outlined in [48]. Hence, we provide the given category with a factorization structure and a special class of morphisms of which we think as of the closed morphisms, satisfying threeF basic axioms; however, there is no a-priori provision of “closure” of subobjects. Depending on the parameter , we introduce and study basic topological properties as mentioned previously, butF encounter also more advanced topics, such as exponentiability. The following features distinguish our presentation from the treatment of the same topological themes in existing topology books: 1. We emphasize the object-morphism interplay: every object notion corresponds to a morphism notion, and vice versa. For example, compact spaces “are” proper 104 III. A Functional Approach to General Topology

maps, and conversely. Consequently, every theorem on compact spaces “is” a theorem on proper maps, and conversely.

locally open separated closed dense ( , 2.1) ( +, 7.1) (2.2) (10.6) F F @ @

initial separated proper ﬁnal surjective (6.5) ( 0, 4.2) ( ∗, 3.1) (8.6) ( , 1.2) F F E @ @@ @ @ @ @ @ @ @ @ @ @

absolutely Tychonoﬀ exponentiable quotient closed (6.1) ( exp, 10.1) (9.1) (6.1) F

locally local perfect homeomorphism (8.1) (7.4)

perfect biquotient

@ ( ∗ 0, 5.2) (10.3) F ∩ F @ @ @

@ @ @ @@ @@ closed open proper stable open stable embedding embedding surjection surjection + + ( 0 = , 2.1) ( , 7.1) ( ∗ ∗, 10.3) ( ∗, 10.3) F F ∩ M F ∩ M F ∩ E F ∩ E Table 1

2. We reach a wide array of applications not only by considering various categories, such as the categories of topological spaces, of locales (see Chapter II), etc., but also by varying the notion of closed morphism within the same category. 105

For example, in our setting, compact spaces behave just like discrete spaces, and perfect maps just like local homeomorphisms, because these notions arise from a common generalization. 3. Our categorical treatment is entirely constructive. In particular, we avoid the use of the Axiom of Choice (also when we interpret the categorical theory in the category of topological spaces), and we restrict ourselves to finitary properties (with the exception of some general remarks on the Tychonoff Theorem and the existence of the Stone-Cechˇ compactification at the end of the chapter). We expect the Reader to be familiar with basic categorical notions, such as adjoint functor and limit, specifically pullback, and we also assume familiarity with basic topological notions, such as topological space, neighbourhood, closure of subsets, continuous map. The Reader will find throughout the chapter mostly easy exercises; those marked with are deemed to be more demanding. ∗

locally Hausdorﬀ (10.6)

-Haus -Comp F(4.3) F (3.3)

@ @ @ @ @ -AC -Tych exponentiable F(6.1) F(6.1) (10.4)

-LCompHaus F (8.1)

-CompHaus F (5.3) Table 2 106 III. A Functional Approach to General Topology

Our basic hypotheses on the category we are working with are formulated as axioms (F0)-(F2) in 1.2 (axioms for factorization systems), (F3)-(F5) in 2.1 (axioms for closed maps), (F6)-(F8) in 11.1 (axioms giving closures), (F9) in 11.2 (inﬁnite product axiom). The morphism notions discussed in this chapter are summarized in Table 1, which also indicates the relevant subsections and the abbreviations used. Upward- directed lines indicate implications; some of these may need extra (technical) hypotheses. Table 2 replicates part of Table 1 at the object level. Many of these notions get discussed throughout the chapter in standard examples, as introduced in Section 2. We list some of them here in summary form, for the Reader’s convenience: op: topological spaces, = closed maps (2.3), T F opopen: topological spaces, = open maps (2.8), T F opclopen: topological spaces, = maps preserving clopen sets (2.6), T F opZariski: topological spaces, = Zariski-closed sets (2.7), T oc: locales, = closedF maps (2.4), LbGrp: abelian groups,F = homomorphisms whose restriction to the A torsion subgroupsF is surjective (2.9), any topos with a universal closure operator (2.5), any ﬁnitely-complete category with a Dikranjan-Giuli closure operator (2.5), any lextensive category with summand-preserving morhisms (2.6), any comma-category of any of the preceding categories (2.10). The authors are grateful for valuable comments received from many colleagues and the anonymous referees which helped us getting this chapter into its current form. We particularly thank Jorge Picado, AleˇsPultr and Peter Johnstone for their interest in resolving the problem of characterizing open maps of locales in terms of closure (see Section 7.3), which led to the comprehensive solution given in [31].

1. Subobjects, images, preimages

1.1. Motivation. A mapping f : X Y of sets may be decomposed as → Z (1) e > @ m }} @@ }} @ X / Y f with a surjection e followed by an injection m, simply by taking Z = f[X], the image of f; e maps like f, and m is an inclusion map. Hence, f = m e with an epimorphism e and a monomorphism m of the category et of sets. If· f is a S 1. Subobjects, images, preimages 107

homomorphism of groups X, Y , then Z is a subgroup of Y , and e and m become epi- and monomorphisms in the category rp of groups, respectively. If f is a continuous mapping of topological spaces XG, Y , then Z may be endowed with the subspace topology inherited from Y , and e and m are now epi- and monomorphisms in the category op of topological spaces, respectively. However, the situation in op is rather diﬀerentT from that in et and rp: applying the decomposition (1) toT a monomorphism f, in et and Srp we obtainG an isomorphism e, but not so in op, simply because theS et-inverseG of the continuous bijective mapping e may failT to be continuous. S In what follows we therefore consider a (potentially quite special) class of monomorphisms and, symmetrically, a (potentially quite special) class Mof epimorphisms in a category and assume the existence of ( , )-decompositionsE for all morphisms, as follows.X E M

1.2. Axioms for factorization systems. Throughout this chapter we work in a ﬁnitely-complete category with two distinguished classes of morphisms and such that X E M (F0) is a class of monomorphisms and is a class of epimorphisms in , Mand both are closed under compositionE with isomorphisms; X (F1) every morphism f decomposes as f = m e with m , e ; (F2) every e is orthogonal to every m · (written∈ M as e ∈m E), that is: given any∈ morphismsE u, v with m u =∈ Mv e, then there⊥ is a uniquely determined morphism w making the· following· diagram commutative:

u / (2) · Ñ@ · e Ñ m Ñ w Ñ / · v · Any pair ( , ) satisfying conditions (F0)-(F2) is referred to as a proper (orthogonal) factorizationE M system of ; “proper” gets dropped if one allows and to be arbitrary morphism classes,X i.e., if one drops the “mono” and “epi”M conditionE in (F0). In that case the unicity requirement for w in (F2) becomes essential; however it is redundant in our situation.

Exercises. 1. Show that every category has two trivial but generally non-proper factorization systems: (Iso, All), (All, Iso).X 2. Show that (Epi, Mono) is a proper factorization system in et and rp, but not in op. S G 3. FindT a class such that (Epi, ) is a proper factorization system in op. 4. Show that f M: X Y is an epimorphismM in the full subcategory aus Tof Hausdorﬀ spaces in op if and→ only if f is dense (i.e., its image meets everyH non-empty open set in Y ).T Find a class such that (Epi, ) is a proper factorization system in aus. Compare this withM op. M H T 108 III. A Functional Approach to General Topology

(Here Iso, Epi, Mono denotes the class of all iso-, epi-, monomorphisms in the respective category, and All the class of all morphisms.)

1.3. One parameter suﬃces. Having introduced factorization systems with two parameters we hasten to point out that one determines the other: Lemma. = f mor m : f m , E { ∈ X | ∀ ∈ M ⊥ } = f mor e : e f . M { ∈ X | ∀ ∈ E ⊥ }

Proof. By (F2), every f satisfies f m for all m . Conversely, if f mor has this property, we may∈ E write f = m⊥ e with e ∈ Mand m , by (F1),∈ andX then apply the hypothesis on f as follows:· ∈ E ∈ M e / (3) · Ñ@ · f Ñ m Ñ w Ñ / · 1 · The monomorphism m with m w = 1 is now recognized as an isomorphism, and we obtain f = m e with (F0).· This shows the first identity, the second follows · ∈ E dually. Exercises. 1. Show that the Lemma holds true without assuming properness of the factorization system. 2. Show that ( , ) is a (proper) factorization system of if and only if ( , ) is a (proper) factorizationE M system of op. Conclude thatX the second identityM ofE the Lemma follows from the first. X 3. Using properness of ( , ), show that if g f = 1, then f and g . More generally: every extremalE M monomorphism (so· that m = h e with∈ Me epic only∈ E if e iso) lies in ; dually, every extremal epimorphism lies in . · M E 1.4. Properties of . We list some stability properties of and : M E M Proposition. (1) and are both closed under composition, and = Iso. (2) IfE n mM , then m . E ∩ M (3) is· stable∈ M under pullback∈ M ; hence, for every pullback diagram M f 0 / (4) · ·

n0 n / · f ·

n implies n0 . ∈ M ∈ M 1. Subobjects, images, preimages 109

(4) is stable under intersection (=multiple pullback); hence, if in the Mcommutative diagram

Mi (5) ji = C mi zz CC zz m C! M / X

(M, m, ji)i I is the limit of the diagram given by the morphisms mi (i I), then∈ also m . ∈ M ∈ ∈ M (5) With all mi : Xi Yi (i I) in , also mi : Xi Yi is in → ∈ M Y Y → Y i I i I i I (if the products exist). ∈ ∈ ∈ M Proof. We show (1), (2) and leave the rest as an exercise (see below). Clearly, if we factor an isomorphism f = m e as in (F1), both e, m are isomorphisms as well, hence f with (F0).· Conversely, f f for f shows that f must be an isomorphism.∈ E ∩ M ⊥ ∈ E ∩ M Given a composite morphism n m in , we may factor m = m0 e with e , · M · ∈ E m0 and obtain from e n m a morphism w with w e = 1. Hence, the ∈ M ⊥ · · epimorphism e is an isomorphism, and m = m0 e . This shows (2). Having again a composite f = n m with both· ∈n, M m , we may now factor · ∈ M f = m∗ e∗ with e∗ , m∗ . Then e∗ n gives a morphism w with w e∗ = · ∈ E ∈ M ⊥ · m , hence e∗ = Iso by what already has been established, and f follows.∈ M ∈ E ∩M ∈ M The assertion on follows dually. E Exercises. 1. Complete the proof of the Proposition, using the same factorization technique as in the part already established. 2. Formulate and prove the dual assertions for ; for example: if e d , then e . 3. For any class of morphisms, let := Ef mor e · ∈: E e f . Show∈ E that is closedE under composition,M and under{ arbitrary∈ X | limits ∀ ∈; thatE is: for⊥ } functors M H,K : and any natural transformation µ : H K with µd for all d , alsoD → X → ∈ M ∈ D lim µ : lim H lim K −→ is in (if the limits exist). Dualize this statement. 4. ProveM that any pullback-stable class satisﬁes: if n m with n monic, then m . M · ∈ M ∈ M 5. ∗ Prove that any class closed under arbitrary limits (see 3) satisﬁes assertions (3)-(5) of the Proposition.M

1.5. Subobjects. We usually refer to morphisms m in as embeddings and call those with codomain X subobjects of X. The class of allM subobjects of X is denoted by subX. 110 III. A Functional Approach to General Topology

It is preordered by

m m0 j : m0 j = m; ≤ ⇔ ∃ · j M / M 0 (6) BB z m BB zz ! }z m0 X Such j is uniquely determined and necessarily belongs to . Furthermore, it is easy to see that M

m m0 & m0 m isomorphism j : m0 j = m; ≤ ≤ ⇔ ∃ · we write m ∼= m0 in this case and think of m and m0 as representing the same subobject of X. Exercises. 1. Let A X be sets. Prove that all injective mappings m : M X with m[M] = A represent⊆ the same subobject of X in et (with = Mono). → S M 2. Prove that m : M X in is an isomorphism if and only if m = 1X . → M ∼ As a consequence of Proposition 1.4 we note:

Corollary. ( , )-factorizations are essentially unique. Hence, if f = m e = m0 e0 E M · · with e, e0 , m, m0 , then there is a unique isomorphism j with j e = e0, ∈ E ∈ M · m0 j = m; in particular, m = m0. · ∼ Proof. Since e m0 there is j with j e = e0, m0 j = m, and with Prop. 1.4(1),(2),(2)op we⊥ have j = Iso.· · ∈ E ∩ M 1.6. Image and preimage. For f : X Y in and m subX, one deﬁnes the image f[m] subY of m under f by an→ ( , X)-factorization∈ of f m, as in ∈ E M · e M / f[M] (7)

m f[m] f X / Y 1 The preimage (or inverse image) f − [n] subX of n subY under f is given by the pullback diagram (cp. 1.4(3)) ∈ ∈

f 1 0 f − [N] / N (8)

1 f − [n] n f X / Y

The pullback f 0 of f along n is also called a restriction of f. Both, image and preimage are uniquely deﬁned, up to isomorphism, and the constructions are re- lated, as follows: 1. Subobjects, images, preimages 111

Lemma. For f : X Y and m subX, n subY one has: → ∈ ∈ 1 f[m] n m f − [n]. ≤ ⇔ ≤

Proof. Consider diagrams (7) and (8). Then, having j : f[M] N with n j = f[m] 1 1 → · gives k : M f − [N] with f − [n] k = m, by the pullback property of (8). Viceversa, the→ existence of k gives the· existence of j since e n. ⊥ Exercises. 1. Conclude from the Lemma: 1 1 (a) m f − [f[m]], f[f − [n]] n; ≤ ≤ 1 1 (b) m m0 f[m] f[m0], n n0 f − [n] f − [n0]. ≤ 1 ⇒ ≤ ≤ ⇒ ≤ 2. Show: f − [1Y ] = 1X , and (f f[1X ] = 1Y ). ∼ ∈ E ⇔ ∼ 1.7. Image is left adjoint to preimage. Lemma 1.6 (with the subsequent Exercise) gives: Proposition. For every f : X Y there is a pair of adjoint functors → 1 f[ ] f − [ ] : subY subX. − a − −→ 1 Consequently, f − [ ] preserves all infima and f[ ] preserves all suprema. − − Exercises. 1. Complete the proof of the Proposition. 2. For g : Y Z, establish natural isomorphisms → 1 1 1 (g f)[ ] = g[ ] f[ ], (g f)− [ ] = f − [ ] g− [ ], · − ∼ − · − · − ∼ − · − 1 1X [ ] = idsubX , 1− [ ] = idsubX . − ∼ X − ∼ 3. Show that the -classes of elements in subX carry the structure of a (possibly large) ∼= meet-semilattice, with the largest element represented by 1X , and with the infimum 1 1 m n represented by m (m− [n]) = n (n− [m]). ∧ · ∼ · 1.8. Pullback stability. We give sufficient conditions for the image-preimage functors to be partially inverse to each other: Proposition. Let f : X Y be a morphism in . → 1 X (1) If f , then f − [f[m]] ∼= m for all m subX. (2) If every∈ M pullback of f along a morphism in∈ lies in (so that in every M E 1 pullback diagram (4) with n one has f 0 ), then f[f − [n]] ∼= n for all n subY . ∈ M ∈ E ∈ Proof. (1) If f , in (7) we may take f[m] = f m and e = 1M , and since f is a monomorphism,∈ M (7) is now a pullback diagram, which· implies the assertion. (2) Under the given hypothesis, in (8) the morphism f 0 lies in , hence n is the 1 E image of f − [n] under f. 112 III. A Functional Approach to General Topology

Exercises. 1. Prove that the suﬃcient condition given in (2) is also necessary. 2. Check that for the factorization systems of et, rp, op given in Exercises 2 and 3 of 1.2, the class = Epi is stable under pullback,S G butT not so for the factorization system of aus describedE in Exercise 4 of 1.2. 3. With the factorizationH system (Epi, ) of op, ﬁnd a morphism f : X Y with 1 M T → f − [f[M]] = M for all M X which fails to belong to . ⊆ M

2. Closed maps, dense maps, standard examples

2.1. Axioms for closed maps. A topology on a set X is traditionally defined by giving a system of open subsets of X which is stable under arbitrary joins (unions) and finite meets (intersections); equivalently, one may start with a system of closed subsets of X which is stable under arbitrary meets and finite joins, with the correspondence between the two approaches given by complementation. Continuity of f : X Y is then characterized by preservation of closed subsets under inverse image.→ Among the continuous maps, these are those for which closedness of subsets is also preserved by image, called closed maps . Since closedness of subobjects is transitive, closed subobjects are precisely those subobjects for which the representing morphism is a closed map. In what follows we assume that in our finitely-complete category with its proper ( , )-factorization system (satisfying (F0)-(F2)) we are givenX a special class EofM morphisms of which we think as of the closed maps, satisfying the followingF conditions: (F3) contains all isomorphisms and is closed under composition; (F4) F is stable under pullback; (F5) wheneverF ∩ M g f with f , then g . · ∈ F ∈ E ∈ F We often refer to the morphisms in as -closed maps, and to those of F F 0 := F F ∩ M as -closed embeddings . Having fixed does not prevent us from considering furtherF classes with the same properties;F hence, we call any class of morphisms in ( , )-closed if satisfies (F3)-(F5) in lieu of . H X E M H F Exercises. 1. Show that the class of closed maps in op (with the factorization structure of Exercise 3 of 1.2) satisfies (F3)-(F5). T 2. A continuous map of topological spaces is open if images of open sets are open. Prove that the class of open morphisms in op is (like the system of closed maps) (Epi, )-closed. T M 3. Prove that each of the following classes in is ( , )-closed: Iso, , , 0. X E M E M F 2. Closed maps, dense maps, standard examples 113

4. Given a pullback-stable class 0 containing all isomorphisms and being closed under composition, deﬁne theH morphism⊆ M class by H (f : X Y ) m subX, m 0 : f[m] 0. → ∈ H ⇔ ∀ ∈ ∈ H ∈ H Show: = 0, and is ( , )-closed if every pullback of a morphism in H ∩ M H H E M E along a morphism in 0 lies in (see Proposition 1.8(2)). Furthermore, satisﬁes the dual of (F5): wheneverH g f E with g , then f . H · ∈ H ∈ M ∈ H 5. Find an ( , )-closed system which does not arise from a class 0 as described in ExerciseE 4.M H H The following statements follow immediately from the axioms (see also Exercise 3 of 1.7). Proposition. (1) The -closed subobjects of an object X form (a possibly large) subsemi- latticeF of the meet-semilattice subX. 1 (2) Every morphism f : X Y in is -continuous, that is: f − [ ] preserves -closedness of→ subobjects.X F − (3) For everyF -closed morphism f : X Y in , f[ ] preserves - closedness ofF subobjects (but not necessarily→ viceversa,X see− Exercise 5)F. 2

2.2. Dense maps. A morphism d : X Y of is -dense if in any factorization d = m h with an -closed subobject→m one necessarilyX F has m Iso. · F ∈ Lemma. d is -dense if and only if d m for all m 0. F ⊥ ∈ F Proof. “if”: From d = m h with m 0 and d m one obtains m iso as in diagram · ∈ F ⊥ 1 (3). “only if”: If v d = m u, one obtains h with d = n h, where n = v− [m] 0 · · · 1 ∈ F by Prop. 2.1. Consequently, n is iso, and we may put w = v0 n− with the pullback 1 1 · v0 of v along m. Then m w = m v0 n− = v n n− = v, which implies w u = d, · · · · · · as desired.

Corollary. (1) Any morphism in is -dense. (2) An -dense and E-closedF subobject is an isomorphism. (3) TheF class of all F -dense morphisms is closed under composition and satisﬁes the dualsF of properties (2)-(5) of Proposition 1.4.

Proof. See Lemma 1.3 and Exercises 1.4.

Exercises.

1. Prove that f : X Y is an -dense morphism if and only if f[1X ]: f[X] Y is an -dense embedding.→ F → 2.F Show that in op (with the class of closed maps, as in Exercise 1 of 2.1) a morphism d : X Y isT -dense ifF and only if every non-empty open set in Y meets d[X]. 3. Verify→ that theF class of -dense maps in op is not (Epi, )-closed. F T M 114 III. A Functional Approach to General Topology

In what follows we shall discuss some important examples which we shall refer to later on as standard examples, by just mentioning the respective category ; the classes , , are understood to be chosen as set out below. X E M F 2.3. Topological spaces with closed maps. The category op will always be considered with its (Epi, )-factorization structure; here Tis the class of embeddings, i.e., of those monomorphismsM m : M X for whichM every closed set of M has 1 → the form m− [F ] for a closed set F of X. (The fact that is precisely the class of regular monomorphisms of the category op is not beingM used in this chapter.) We emphasize again that here = Epi is stableT under pullback (see Exercise 2 of 1.8). In the standard situation,E is always the class of closed maps. -closedness and -density for subobjects takeF on the usual meaning. F F 2.4. Locales with closed maps. The category oc of locales has been introduced as the dual of the category rm of frames in ChapterL II. The (Epi, )-factorization structure of oc coincidesF with the ( , Mono)-factorization structureM of rm, where is theL class of surjective frameE homomorphisms; following the notationF of ChapterE II, this means:

m : M X in m∗ : X M in . → M ⇔ O → O E The subobject m is closed if there is a X and an isomorphism h : M a ∈ O O →↑ of frames such that h m∗ =a ˇ, witha ˇ : X a given bya ˇ(x) = a x. The class of closed maps· in oc can now beO defined→↑ just as in op, by preservation∨ of closedF subobjects underL image (see II.5.1). T We point out a major difference to the situation in op: the class Epi in oc fails to be stable under pullback (see II.5.2). However, pullbacksT of epimorphismsL along complemented sublocales (which include both, closed sublocales and open sublocales – see II.2.7) are again epic in oc, since for any morphism f in oc the 1 L L inverse image functor f − [ ] preserves finite suprema, and, hence, also complements (see II.3.8). − Next we mention three (schemes of) examples of a more general nature. 2.5. A topos with closed maps with respect to a universal closure operator. Every elementary topos has an (Epi, Mono)-factorization system, and the class Epi is S stable under pullback (see [28], 1.5). A universal closure operator j = (jX )X ob is ∈ S given by functions jX : subX subX satisfying the conditions 1. m jX (m), 2. → 1 ≤ 1 m m0 jX (m) jX (m0), 3. jX (jX (m)) = jX (m), 4. f − [jY (n)] = jX (f − [n]), ≤ ⇒ ≤ ∼ ∼ for all f : X Y , m, m0 subX, n subY (see [28], 3.13, and [36], V.1). Taking → ∈ ∈ 1 for now the class of those morphisms f for which not only f − [ ] but also f[ ] commutesF with the closure operator, one obtains an (Epi, Mono)-closed− system− for ; its closed maps are also described by the fact that image preserves closednessF S of subobjects (characterized by m ∼= jX (m)). A very particular feature of this structure is that the class of -dense maps is stable under pullback; see Chapter VII. F Although we shall not discuss this aspect any further in this chapter, we mention that the notion of universal closure operator may be generalized dramatically 2. Closed maps, dense maps, standard examples 115

if we are just interested in the generation of a closed system. In fact, for any finitely-complete category with a proper ( , )-factorization system and a so- called Dikranjan-Giuli closureX operator one mayE M take for those morphisms for which image commutes with closure (see [18, 9] for details,F as well as Section 11 below). 2.6. An extensive category with summand-preserving morphisms. A finitely-complete category with binary coproducts (denoted by X +Y ) is (finitely) extensive if the functor X pb : /(X + Y ) /X /Y X,Y X −→ X × X (given by pullback along the coproduct injections) is an equivalence of categories, for all objects X and Y ; equivalently, one may ask its left adjoint (given by coproduct) to be an equivalence, see [6, 5, 7]. (Note that in Chapter VII extensive is used in the infinitary sense where the binary coproducts are replaced by arbitrary ones, see VII.4.1; however, in this chapter extensive always means finitely extensive.) In order to accommodate our setting we also assume that has a proper ( , )-factorization structure such that every coproduct injectionX lies in . Tak- E M M ing for 0 now those morphisms m : M X for which there is some morphism n : N F X such that X = M + N with→ coproduct injections m, n, one lets → ∼ F contain those morphisms for which image preserves the class 0: see Exercise 4 of 2.1. F For example, = op is extensive, and the -closed morphisms are precisely those maps whichX mapT clopen (=closed and open)F sets onto clopen sets. When- ever we refer to op with this structure and its (Epi, )-factorization system, T F M we write opclopen instead of op. Another prominent example of an extensive category isT the dual of the categoryT ng of commutative rings (where ng is considered with its ( , Mono)-factorizationCR system); for details we refer toCR [5]. E Exercises. 1. Prove that the finitely complete category with binary coproducts is extensive if and only if in every commutative diagram X A / C o B (9)

i1 i2 X / X + Y o Y the top row represents a coproduct (so that C A + B) precisely when the two ∼= squares are pullback diagrams. 2. Prove that it is enough to require the existence of the coproduct 1 + 1 and that

pb1,1 is an equivalence to obtain extensivity, including the existence of all binary coproducts. 3. Prove that in an extensive category coproduct injections are monomorphisms, and that they are disjoint, i.e. the pullback of i1, i2 is an initial object. In particular, has an initial object. X 116 III. A Functional Approach to General Topology

4. Prove that an extensive category is distributive, so that the canonical morphism (X Y ) + (X Y ) X (Y + Y ) is an isomorphism. In addition, X 0 0, 1 2 1 2 ∼= with× 0 denoting× the initial→ object× (see Ex. 3). × 5. Verify the extensivity of op and ngop, and of every topos. T CR

ni 2.7. Algebras with Zariski-closed maps. For a fixed set K let Ω = ωi : K { → K i I be a given class of operations on K of arbitrary arities; hence, the ni are arbitrary| ∈ } cardinal numbers, and there is no condition on the size of the indexing system I. The category Ω- et has as objects sets X which come equipped with an Ω-subalgebra A(X) of KSX ; here KX carries the pointwise structure of an Ω- algebra: (KX )ni = (Kni )X KX . ∼ → A morphism f : X Y in Ω- et must satisfy Kf [A(Y )] A(X), that is: f β A(X) for all β →A(Y ). LikeS op, Ω- et has a proper⊆ (Epi, )-factorization· ∈ structure, with the∈ morphisms inT representedS by embeddings MM, X, so that M → A(M) = α M α A(X) . Such a subobject M is called Zariski-closed if any x X with{ | | ∈ } ∈ α, β A(X)(α M = β M α(x) = β(x)) ∀ ∈ | | ⇒ already lies in M. The class of closed maps contains precisely those maps for which image preserves Zariski-closednessF (see [15, 22]). We mention in particular two special cases. First, take K = 2 = 0, 1 , with the operations given by the frame structure of 0 1 , that is: by arbitrary{ } joins and finite meets. Equipping a set X with an Ω-subalgebra{ ≤ } of 2X is putting a topology (of open sets) on X, and we have Ω- et = op. The Zariski-closed sets on X define a new topology on X with respectS toT which a basic neighbourhood of a point x X has the form U x , where U is a neighbourhood of x and x the closure of∈ x in the original∩ topology.{ } Whenever we refer to op with given{ } as { } T F above, we shall write opZariski. Secondly, in the “classical”T situation, one considers the ground field K in the category of commutative K-algebras, so the operations in Ω are 0, 1 : K0 K, a( ): K K, +, : K2 K, → − → · → where a( ) is (left) multiplication by a, for every a K. Hence, an Ω-set X comes with a subalgebra− A(X) of KX , and M X is Zariski-closed∈ if ⊆ M = x X α A(X)(α M = 0 α(x) = 0) . { ∈ | ∀ ∈ | ⇒ } Putting J = α A(X) α M = 0 one sees that the Zariski-closed sets are all of the form { ∈ | | } Z(J) = x X α J : α(x) = 0 { ∈ | ∀ ∈ } for some ideal J of A(X). For X = Kn and A(X) the algebra of polynomial functions Kn K we have exactly the classical Zariski-closed sets considered as in commutative→ algebra and algebraic geometry (see [15]). 2. Closed maps, dense maps, standard examples 117

Exercises.

1. Let X be the cartesian product of the sets Xν , with projections pν (ν N). If X ∈ each Xν is an Ω-set, let A(X) be the subalgebra of K generated by α pν ν { · | ∈ N, α A(Xν ) . Conclude that Ω- et has products. More generally, show that Ω- et is small-complete.∈ } S S 2. Show that in opZariski every Zariski-closed subset of a subspace Y of X is the intersection ofT a Zariski-closed set in X with Y . Generalize this statement to Ω- et. extS 3. Consider K = 2 = 0, 1 and Ω = . Show that Ω- et is the category hu2 of extensive Boolean Chu{ spaces} (see [42]),∅ whose objectsS are sets X with a “generalizedC topology” τ 2X (no further condition on τ), and whose morphisms are “continuous maps”. Prove⊆ that a subspace M of X is Zariski-closed if for every x X M there are (“open sets”) U, V τ with U M = V M and x U V . ∈ \ ∈ ∩ ∩ ∈ \

2.8. Topological spaces with open maps. Here we mention a somewhat counter- intuitive but nevertheless important example. Again, we consider op with its (Epi, )-factorization system, but now take to be the class of openT maps (see ExerciseM 2 of 2.1). Here the -dense maps fF: X Y are characterized by the F → property that the closure y of every point y Y meets the image f[X]. When { } ∈ equipping op with this structure, we refer to opopen rather than to op (see 2.3). T T T

2.9. Abelian groups with torsion-preserving maps. The category bGrp of abelian groups has, like rp, an (Epi, Mono)-factorization system. Here weA consider for the class of homomorphismsG f : A B which map the torsion subgroup of FA onto the torsion subgroup of B: f[Tor→ A] = TorB = b B nb = 0 for some n 1 . (More generally, for any ring R, we could consider{ ∈ here| any preradical of R-modules≥ } in lieu of Tor, or even of any ﬁnitely-complete category with a proper factorization system, in lieu of bGrp; see [18].) The map f is -dense if and only if B = imf + TorB. A F We end our list of standard examples with an important general procedure:

2.10. Passing to comma categories. Given our standard setting with , , , satisfying (F0)-(F5) and a ﬁxed object B in , let /B be the commaX E categoryM F (or sliced category) of objects (X, p) over B, simplyX givenX by morphisms p : X B in ; a morphism f :(X, p) (Y, q) in /B is a morphism f : X Y in→ X → X → 1 X with q f = p. With the forgetful functor ΣB : /B , putting B := ΣB− ( ), · 1 1 X → X E E B := ΣB− ( ), B := ΣB− ( ), it is easy to check that (F0)-(F5) hold true in M/B. M F F X Proposition. For every object B of , ( B, B) is a proper factorization system X E M of /B and B is an ( B, B)-closed class. 2 X F E M In what follows, whenever we refer to /B, we think of it as being structured X according to the Proposition. This applies particularly to op/B, opopen/B, etc. T T We normally drop the subscript B when referring to B, B, B. E M F 118 III. A Functional Approach to General Topology

3. Proper maps, compact spaces

3.1. Compact objects. The pullback of a closed map in op need not be a closed map. In fact, the only map from the real line R to a one-pointT space 1 is closed, but the pullback of this map along itself is not: R R / R (10) ×

R / 1 neither projection of R R preserves closedness of the subspace (x, y) x y = 1 . Hence, we should pay× attention to those closed maps which are{ stable| under· pullback.} In our ﬁnitely-complete category with its proper ( , )-factorization system and the distinguished class ofX closed maps (satisfyingE M (F0)-(F5)) we call a morphism f -proper if f belongsF stably to , so that in every pullback diagram F F f 0 W / Z (11)

g0 g f X / Y f 0 ; laxly we speak of an -proper map in and denote by ∗ the class of all ∈ F F X F -proper maps. Note that since g and g0 may be chosen as identity morphisms, F ∗ . F Checking⊆ F -properness of a morphism may be facilitated by the following criterion: F Proposition. A morphism f : X Y is -proper if and only if every restriction → F of f 1Z : X Z Y Z (see (8)) is -closed, for every object Z. × × → × F Proof. Since with f 1Z also every of its restrictions is a pullback of f, the necessity of the condition× is clear. That it is suﬃcient for -properness follows from a factorization of the pullback diagram (11), as follows:F

f 0 W / Z (12)

f 1Z X Z × / Y Z × × p1 p1 f X / Y Since both the total diagram and its lower rectangle are pullback diagrams, so is its upper rectangle. Moreover, n :=< g, 1Z > (see Exercise 3 of 1.3), so that ∈ M the condition of the Proposition applies here. 3. Proper maps, compact spaces 119

Often, but not always, the condition of the Proposition can be simpliﬁed. Let us say that is stable under restriction if every restriction of an -closed morphism is -closed,F i.e., if is stable under pullback along morphismsF in . In this case F F M f is -proper if and only if f 1Z is -closed for all Z. F × F Exercises.

1. Show that in op, opopen, opclopen, opZariski, the class is stable under restriction. T T T T F 2. Using = , show that generally fails to be stable under restriction. 3. Let F= Ewith as in Exercise 4F of 2.1, and assume that is stable under pullback alongF H, and thatH every -closed subobject of M with mE : M X in is the inverseM image of an -closedF subobject of X under m. Show that→ is stableM under restriction. Revisit ExerciseF 1. F

3.2. Stability properties. We show some important stability properties for - proper maps: F Proposition.

(1) The class ∗ contains and is closed under composition. F F ∩ M (2) ∗ is the largest pullback-stable subclass of . F F (3) If g f ∗ with g monic, then f ∗. · ∈ F ∈ F (4) If g f ∗ with f ∗, then g ∗; here ∗ is the class of the morphisms· ∈ F stably in .∈ E ∈ F E E Proof. (1) follows from (F4) and the fact that a pullback of g f can be obtained as a composite of a pullback of g preceded by a pullback of f. Likewise,· (2) and (4) follow from the composability of adjacent pullback diagrams. (3) is a consequence of Exercise 4 of 1.4.

Exercises. 1. Make sure that you understand every detail of the proof of the Proposition. 2. Show that with f : X Y also the following morphisms are -proper: (a) f m : M Y for→ every -closed embedding m : M XF; · 1 → F → (b) f 0 : f − [N] N for every n subY . → ∈ 3. Using only closure of ∗ under composition and pullback stability, show that with F f1 : X1 Y1 and f2 : X2 Y2 also f1 f2 : X1 X2 Y1 Y2 is -proper. (We → → × × → × F say that ∗ is closed under ﬁnite products. Hint: f1 f2 = (f1 1) (1 f2).) F × × · × 4. Show that in the standard example op (see 2.3) one has ∗ = ; likewise for T E E opopen, opclopen, opZariski, and for every topos, but not for aus and oc. T T T H L

3.3. Compact objects. An object X of is called -compact if the unique mor- X F phism !X : X 1 to the terminal object is -proper. Since the pullbacks of !X are the projections→X Y Y , Y , to sayF that X is -compact means precisely that any such projection× → must be∈ X in . Before exhibitingF this notion in terms of examples, let us draw some immediateF conclusions from Prop. 3.2: 120 III. A Functional Approach to General Topology

Theorem. (1) For any -proper f : X Y with Y -compact, X is -compact. F → F F (2) For any f : X Y in ∗ with X -compact, Y is -compact. (3) The full subcategory→ -EComp of F-compact objectsF in is closed under ﬁnite products and underF -closedF subobjects. X F Proof. (1) Since

f !Y !X = (X Y 1), −→ −→ the assertion follows from the compositivity of ∗ (Prop. 3.2(1)). (2) For the same reason one may apply Prop. 3.2(4)F here. (3) Since ∗ contains all isomorphisms one has 1 -Comp, and for X,Y - Comp oneF can apply (1) to the projection X Y ∈ FY . Likewise, one can apply∈ F × → (1) to f 0 to obtain closure under -closed subobjects. ∈ F F Exercises. 1. Show that every -closed subobject of an -compact object is -compact. 2. Find an exampleF in op of a morphism f : FX Y with X,Y F -Comp which fails to be -proper. T → ∈ F 3. ShowF that -Comp in op fails to be closed under finite limits. F T 3.4. Categorical compactness in op. We show that in op (with = closed maps ) our notion of -compactnessT coincides with the usualT notionF of compact-{ ness given} by the Heine-BorelF open-cover property: Theorem (Kuratowski and Mrówka). For a topological space X, the following are equivalent: (i) X is -compact, i.e., the projection X Y Y is closed for all Y op; F × → ∈ T (ii) for every family of open sets Ui X (i I) with X = i I Ui, there is ⊆ ∈ S ∈ a finite set F I with X = i F Ui. ⊆ S ∈ Proof. (ii) (i): This part is the well-known Kuratowski Theorem, we only sketch ⇒ a possible proof here (see also Exercise 1 below). To see that B = p2(A) is closed for A X Y closed and p2 : X Y Y the second projection, assume y B B. ⊆ × 1 × → ∈ \ Then A p− [V ] V neighbourhood of y is a base of a filter F on X Y . The { ∩ 2 | } × filter p1(F) on X must have a cluster point x, hence 1 x p1[A p− [V ]] V neighbourhood of y , ∈ \{ ∩ 2 | } which implies (x, y) A A: contradiction. ∈ \ (i) (ii): With the given open cover Ui i I of X, define a topological space Y with⇒ underlying set X (with{ | X∈) by} ∪ {∞} ∞ 6∈ K Y closed : K or K Ui for some finite F I. ⊆ ⇔ ∞ ∈ ⊆ [ ⊆ i F ∈ It now suffices to show that X Y is closed in Y , and for that it suffices to prove ⊆ p2(∆X ) = X, with the closure of ∆X = (x, x) x X formed in X Y . But the { | ∈ } × 3. Proper maps, compact spaces 121

assumption p2(∆X ) would give an x X with (x, ) ∆X . Since x lies in ∞ ∈ ∈ ∞ ∈ some Ui we would then have (x, ) Ui (Y Ui), an open set in X Y which ∞ ∈ × \ × does not meet ∆X , a contradiction! We note that the proof of the Theorem does not require the Axiom of Choice. This applies also to the following exercise. Exercise. Extending and refining the argumentation in the proof of the Theorem, prove the equivalence of the following statements for a topological space X: (i) X is -compact; F (ii) p2 : X Y Y is closed for every zerodimensional, normal Hausdorff space Y ; × → (iii) X has the Heine-Borel open-cover property; (iv) every filter F on X has a cluster point, i.e. F F F = . T{ | ∈ } 6 ∅ 3.5. Fibres of proper maps. In 3.3 we defined -compactness via -properness. We could have proceeded conversely, using the followingF fact: F Proposition. f : X Y is -proper if and only if (X, f) is an -compact object of /Y (see 2.10). → F F X Proof. The terminal object in /Y is (Y, 1Y ), and the unique morphism (X, f) X → (Y, 1Y ) is f itself. The question remains whether -properness can be characterized by -compactness within the category . TowardsF this we first note: F X Corollary. For the following statements on f : X Y in , one has (i) (ii) (iii): → F ⇒ ⇒ (i) f is -proper; (ii) for everyF pullback diagram (11), if Z is -compact, so is W ; (iii) all fibres of f are -compact, where a fibreF F of f occurs in any pullback diagram F F / 1 (13)

f X / Y

Proof. (i) (ii): Since ∗ is pullback-stable, in (11) with f also f 0 is -proper. Hence, the⇒ assertion of (ii)F follows from Theorem 3.3(1). F (ii) (iii) follows by putting Z = 1, which is -compact by Theorem 3.3(3). ⇒ F We now show (choice free) that in op condition (iii) is already sufficient for properness: T Theorem. In op a map is -proper if and only if it is closed and has compact fibres. T F Proof. Let f : X Y be closed with compact fibres, and for any space Z let → A X Z be closed; we must show that B := (f 1Z )[A] Y Z is closed. ⊆ × × ⊆ × 122 III. A Functional Approach to General Topology

Hence, for any point (y, z) in the complement (Y Z) B we must find open × \ sets V0 Y , W0 Z with (y, z) V0 W0 (Y Z) B. The system of all ⊆ ⊆ ∈ × ⊆ × \ 1 pairs (U, W ) where U is an open neighbourhood in X of some x f − y and W is an open neighbourhood in Z of z with U W (X Z) A has∈ the property × ⊆ 1 × \ that its first components cover the compact fibre f − y. Hence, we obtain finitely 1 n many open sets U1, ,Un, W1, ,Wn (n 0) with f − y U0 := i=1 Ui, n ··· ··· ≥ ⊆ S z W0 := Wi and U0 W0 (X Z) A. Since f is closed, the set ∈ Ti=1 × ⊆ × \ V0 := Y f(X U0) is open and has the required properties. \ \ However, in general, in op/B not even condition (ii) is sufficient for - properness, as the following exampleT shows: F Example. Let B be any indiscrete topological space with at least 2 points, let X be any non-compact topological space, and let q : 1 B be any map. Then, with p = q f, the unique map f : X 1 becomes a morphism→ f :(X, p) (1, q) in op/B· which is -closed but→ not -proper. However, condition (ii)→ of the CorollaryT is vacuouslyF satisfied since, givenF any pullback diagram (11) (with Y = 1) in which then represents a pullback diagram also in /B, we observe that the objectX (Z, q g) is never -compact in op/B unless Z isX empty; indeed, for Z = the constant· map q g isF not proper, sinceT its image is not closed in B. 6 ∅ · Exercises.

1. In opopen, prove that every map in = f f open is -proper. Conclude that everyT object is -compact. F { | } F F 2. ∗ Prove: a space X in opZariski is -compact if and only if if it is compact with respect to its Zariski topologyT (see 2.7).F Conclude that this is the case precisely when (a) every subspace of X is compact (in op) and (b) every closed subspace of X has T the form x1 xn for finitely many points x1, , xn in X, n 0. { } ∪ · · · ∪ { } ··· ≥ 3.6. Proper maps of locales. In oc, -proper maps are characterized by: L F Theorem (Vermeulen [49, 50]). For f : X Y in oc, the following are equivalent: → L (i) f is -proper; F (ii) f 1Z : X Z Y Z is -closed for all locales Z; (iii) the× restriction× f→[ ]: ×X FY of the direct-image map to closed sublocales is well defined− andC preserves→ C filtered infima. For the proof of this theorem we must refer to Vermeulen’s papers [49, 50]. However, we may point out that the equivalence proof for (i), (ii) given in [50] makes use of a refined version of Exercise 3 of 3.1. We leave it as an exercise to show that when exploiting property (iii) in case Y = 1, one obtains (see Theorem II.6.5): Corollary (Pultr-Tozzi [43]). A locale X is -compact if and only if every open cover of X contains a finite subcover. F

3.7. Sums of proper maps and compact objects. Next we want to examine the behaviour of -compact objects and -proper maps under the formation of ﬁnite F F 3. Proper maps, compact spaces 123

coproducts. Clearly, some compatibility between pullbacks and finite coproducts has to be in place in order to establish any properties, in addition to closedness of under finite coproducts, so that with f1 : X1 Y1, f2 : X2 Y2 also F → → f1 + f2 : X1 + X2 Y1 + Y2 is in . The notion of extensive category (as given in 2.6) fits our requirements:→ F Proposition. Assume that the finitely-complete category has finite coproducts and is extensive, and that is closed under finite coproducts.X Then: F (1) the morphism class ∗ is closed under finite coproducts; (2) the subcategory -CompF is closed under finite coproducts in if and only if the canonicalF morphisms X 0 X and X + X X −→ −→ are -closed, for all objects X. F Proof. (1) Diagram (14) shows how one may obtain the pullback f 0 of f = f1 + f2 along h: pulling back h along the injections of Y = Y1 + Y2

W1 / W o W2 (14) h10 h zz 0 }} zz z f10 } f 0 z f20 |z ~} |z h20 X1 / X o X2 f1 f f2 Z1 / Z o Z2 zz }} zz |zz h1 ~}} h |zz h2 Y1 / Y o Y2

one obtains h1 and h2, along which one pulls back f1 and f2 to obtain f10 and f20 and the induced arrows W1 W W2 giving the commutative back faces. These are pullback diagrams since→ all other← vertical faces are; by extensivity then, f 0 = f 0 + f 0 . Hence with f1, f2 ∗ we have f 0 , f 0 and then f 0 , so that 1 2 ∈ F 1 2 ∈ F ∈ F f ∗. (2)∈ Once F we have 1 + 1 -Comp ∈ F X / X + Y o Y (15)

!X +!Y 1 / 1 + 1 o 1

we obtain X +Y -Comp whenever X,Y -Comp since !X+Y =!1+1 (!X +!Y ), by an application∈ of F (1). Now, ∈ F · 1 + 1 -Comp X : ((1 + 1) X X) ∈ F ⇔ ∀X :(X + X × X→) ,∈ F ⇔ ∀ → ∈ F since (1 + 1) X = X + X (see Exercise 4 of 2.6). Furthermore, × ∼ 0 -Comp X : (0 X X) ∈ F ⇔ ∀X : (0 × X→) ,∈ F ⇔ ∀ → ∈ F 124 III. A Functional Approach to General Topology

since 0 X = 0 (see Exercise 4 of 2.6). × ∼ Exercises.

1. Check that the hypotheses of the Theorem are satisfied in op, opopen, opclopen, T T T opZariski, and in any extensive category with as in 2.6. Conclude that in these Tcategories -Comp is closed under finite coproducts.F 2. Show thatF in = ( ng)op with as in 2.6, the only -compact rings (up to isomorphism) areXZ andCR0 , and thatF -Comp fails to be closedF under finite coproducts in . { } F X

4. Separated maps, Hausdorﬀ spaces

4.1. Separated morphisms. With every morphism f in our category , structured by , , as in 2.1, we may associate the morphism X E M F δf :< 1X , 1X >: X X Y X −→ × where X Y X belongs to the pullback diagram × f2 X Y X / X (16) ×

f1 f f X / Y representing the kernel pair of f. It is easy to see that

δf f1 X / X Y X / X (17) × f2 is an equalizer diagram. Instead of asking whether f ∗, in this section we ∈ F investigate those f with δf ∗. Actually, since δf and 0 = is ∈ F ∈ M F F ∩ M stable under pullback, it is enough to require δf . ∈ F We call a morphism f in -separated if δf ; laxly we speak of an - XF ∈ F F separated map and denote by 0 the class of all -separated maps. F F Example. In op, to say that ∆X = (x, x) x X is closed in the subspace T { | ∈ } X Y X = (x1, x2) f(x1) = f(x2) of X X is to say that (X Y X) ∆X is × { | } × × \ open, and this means that for all x1, x2 X with f(x1) = f(x2), x1 = x2 there ∈ 6 are open neighbourhoods U1 x1, U2 x2 in X with U1 Y U2 (X Y X) ∆X , 3 3 × ⊆ × \ i.e., U1 U2 = . Hence, in op, a map f : X Y is -separated if and only if distinct∩ points6 in∅ the same ﬁbreT of f may be separated→ byF disjoint neighbourhoods in X. Such maps are called separated in ﬁbred topology.

Exercises. Prove:

1. f : X Y is -separated in opopen if and only if f is locally injective, so that for → F T every point x in X there is a neighbourhood U of x such that f U : U Y is an injective map. | → 4. Separated maps, Hausdorﬀ spaces 125

2. The -separated maps in opclopen are precisely the separated and locally injective maps.F Show that neither ofT the latter two properties implies the other. 3. f : X Y is -separated in opZariski if and only if distinct points in the same ﬁbre have distinct→ neighbourhoodF T ﬁlters. 4. f : A B is -separated in bGrp if and only if the restriction of f to TorA is → F A injective, i.e., ker f TorA = 0. |

4.2. Properties of separated maps. When establishing stability properties for 0, F it is important to keep in mind that 0 depends only on 0 = ; hence, only F F F ∩ M the behaviour of 0 plays a role in what follows. F Proposition.

(1) The class 0 contains all monomorphisms of and is closed under composition. F X (2) 0 is stable under pullback. F (3) Whenever g f 0, then f 0. · ∈ F ∈ F (4) Whenever g f 0 with f ∗, then g 0. · ∈ F ∈ E ∩ F ∈ F Proof. (1) Monomorphisms are characterized as those morphisms f with δf iso. Let us now consider f : X Y , g : Y Z and their composite h = g f. Then → → · there is a unique morphism t : X Y X X Z X with h1 t = f1, h2 t = f2. It makes the following diagram commutative:× → × · ·

δf f f1 X / X Y X · / T (18) ×

1X t δg

δh f f X / X Z X × / Y Z Y × × The right square is in fact a pullback diagram since t is the equalizer of f h1 = f h2 · · (with h1, h2 the kernelpair of h). Consequently, with δg also t is in , and then F with δf also δh = t δf is in . (2) · F

δ f0 f 0 W / W Z W / W / Z (19) × /

k0 3 k00 2 k0 1 k δf f X / X Y X / X / Y × / Since with 1 also 2 & 1 and then 3 are pullback diagrams, the assertion follows from the pullback stability of . F ∩ M (3) Going back to (18), if δh = t δf is in the pullback-stable class 0, so is δf since t is monic (see Exercise 4 of· 1.4). F (4) Since f = f f1 δf , from (18) one has δg f = (f f) δh. Now, if h 0 · · · × · ∈ F and f ∗, then (f f) δh = δg f (see Exercise 1). Hence, if also f , ∈ F × · · ∈ F ∈ E δf follows with (F5). ∈ F 126 III. A Functional Approach to General Topology

Exercises.

1. Show that with f1, f2 0 also f1 f2 0 (see Exercise 3 of 3.2). ∈ F × ∈ F 2. In generalization of 1, prove that 0 is closed under those limits under which F F ∩ M is closed. Hint: In the setting of Exercise 3 of 1.4, prove the formula δlim µ = limd δµd .

4.3. Separated objects. An object X of is called -separated (or -Hausdorﬀ) X F F if the unique morphism !X : X 1 is -separated; this simply means that → F δX =< 1X , 1X >: X X X must be in . → × F Theorem. The following conditions are equivalent for an object X: (i) X is -separated; (ii) everyF morphism f : X Y is -separated; (iii) there is an -separated→ morphismF f : X Y with Y -separated; (iv) for every objectF Y the projection X Y → Y is -separated;F (v) for every -separated object Y , X ×Y is→ -separated;F (vi) for every F-proper morphism f : X× Y inF , Y is -separated; (vii) in every equalizerF diagram → E F

u E / Z / X, u is -closed. F

Proof. Since !X =!Y f, the equivalence of (i), (ii), (iii) follows from compositivity · and left cancellation of 0 (Prop. 4.2(1),(3)). Since the projection X Y Y is a F × → pullback of X 1, (i) (iv) follows from pullback stability of 0 (Prop. 4.2(2)), and (iv) (v)→ from its⇒ compositivity again. For (v) (i) considerF Y = 1, which ⇒ ⇒ is trivially -separated. For (i) (vi) apply Prop. 4.2(4) with g =!Y , and for F ⇒ (vi) (i) let f = 1X . Finally (i) (vii) follows since such equalizers u are ⇒ ⇔ precisely the pullbacks of δX .

Corollary. The full subcategory -Haus of -separated objects is closed under ﬁnite limits and under subobjects in F . In fact,F for every monomorphism m : X Y , with Y also X is -separated. X → F Proof. Since every monomorphism is -separated, consider (iii), (iv) of the The- F orem.

Exercises. 1. In op and oc, -separation yields the usual notion of Hausdorff separation of these categories.T L F 2. Show that in opopen and in opclopen the -separated objects are the discrete spaces. ConcludeT that, in general,T -Haus failsF to be closed under (infinite) products in . F X 3. Prove that in opZariski -Haus is the category of T0-spaces (i.e., those spaces in which distinctT points haveF distinct neighbourhood filters). 4. Separated maps, Hausdorff spaces 127

4. The quasi-component qX (M) of a subset M of a topological space X is the intersection of all clopen subsets of X containing M. Let T opquasicomp denote the category op with its (Epi, )-factorization structure and take for the class all q-preserving T M F maps f : X Y , i.e., f[qX (M)] = qY (f[M]). Show that -Haus in opquasicomp is the category→ of totally-disconnected spaces (i.e., spaces inF which all componentsT are singletons). 5. Prove that in bGrp -Haus is the category of torsion-free abelian groups (i.e., groups in whichAna = 0F implies n = 0 or a = 0).

4.4. Sums of separated maps and objects. Closure of 0 and of -Haus under finite F F coproducts requires not only a stability property of 0 under finite coproducts but, like in 3.7, also extensivity of : F X Proposition. Assume that the finitely-complete category has finite coproducts and is extensive, and that the coproduct of two -closedX subobjects is -closed. Then: F F

(1) the morphism class 0 is closed under ﬁnite coproducts; (2) the subcategory -HausF is closed under ﬁnite coproducts in if and only if 1 + 1 -HausF . X ∈ F

Proof. (1) For f1 : X1 Y1, f2 : X2 Y2, by extensivity, as a coproduct of two pullback squares also the→ right vertical→ face of (20) is a pullback square.

X1 Y X1 / (X1 Y X1) + (X2 Y X2) (20) × 1 × 1 × 2 π1 π1+π2 1 1 1 X1 / X1 + X2

1 1 2 π2 π2 +π2

X1 / X1 + X2 f1 f1 f1+f2 f1+f2 Y1 / Y1 + Y2 Hence,

(X1 Y X1) + (X2 Y X2) = (X1 + X2) Y +Y (X1 + X2), × 1 × 2 ∼ × 1 2 and the formula δf +f = δf + δf follows immediately. Consequently, f1, f2 0 1 2 ∼ 1 2 ∈ F implies f1 + f2 0. ∈ F 1 2 1 2 (2) We saw in (1) that (π1 +π1, π2 +π2) is the kernelpair of f1 +f2. Exploiting this fact in case Y1 = Y2 = 1, we see that the canonical morphism u, making diagram 128 III. A Functional Approach to General Topology

(21) commutative, is actually an equalizer of (f1 + f2) π1,(f1 + f2) π2. · · 1 X1 + X2 / X1 + X2 (21)

δX1 +δX2 δX1+X2 u (X1 X1) + (X2 X2) / (X1 + X2) (X1 + X2) × × × 1 2 1 2 π π π1 +π1 π2 +π2 1 2 1 X1 + X2 / X1 + X2 Hence, if 1 + 1 is -separated, then u is -closed (see (vii) of Theorem 4.3), and F F with δX , δX we obtain δX +X . Since 0 0 = 0, trivially δ0 . 1 2 ∈ F 1 2 ∈ F × ∼ ∈ F Exercises.

1. Show that, in op, opopen, opclopen and opZariski, 0 and -Haus are closed under finite coproducts.T T LikewiseT in bGrp. T F F 2. Show that, in general, 1+1 fails to beA -separated. (Consider, for example, = Iso.) F F 4.5. Separated objects in the slices. We have presented -separation for objects as a special case of the morphism notion. We could haveF proceeded conversely, using: Proposition. f : X Y is -separated if and only if (X, f) is an -separated object of /Y (see 2.10)→ . F F X Proof. δf : X X Y X in serves as the morphism δ(X,f) :(X, f) (X, f) (X, f) in /Y→. × X → × X Exercises. 1. Prove Corollary 4.3 without recourse to the previous Theorem, just using the definition of -Hausdorff object. Then apply the assertions of the Corollary to /Y in F X lieu of and express the assertions in terms of properties of ∗. 2. Prove thatX with also each slice /Y is extensive. (Hint: SlicesF of slices of are slices of .) X X X X We finally mention three important “classical” examples, without proofs. Examples. (1) Since commutative unital R-algebras A are completely described by morphisms R A in ng, the opposite of the category lgR is extensive, by Exercise→ 2: CR CA lgop = ngop/R. CA R ∼ CR An -separated object in this extensive category is precisely a separable R-algebra;F see [3]. (2) In a topos, considered as an extensive category (see Exercise 5 of 2.6), -separated means decidable (see [2], p.444). To say that every object is decidableF is equivalent to the topos being Boolean. 5. Perfect maps, compact Hausdorff spaces 129

(3) An object in a topos provided with a universal closure operator j (see 2.5) is -separated precisely when it is j-separated in the usual sense (see [36], p.227).F

5. Perfect maps, compact Hausdorﬀ spaces

5.1. Maps with compact domain and separated codomain. There is an easy but fundamental property which links compactness and separation, in the general setting provided by 2.1: Proposition. If X is -compact and Y -separated, then every morphism f : X Y is -proper. F F → F Proof. One factors f through its “graph”: X Y (22) <1X ,f> u: × IIp2 uu II uu I X /$ Y f

1 Since < 1X , f >= f − [δY ], this morphism is -closed, in fact -proper, when Y ∼ F F is -separated. Likewise p2 is -closed, even -proper, when X is -compact. F F F F Closure of ∗ under composition shows that f is -proper. F F Corollary. If Y is -separated and -compact, then f : X Y is -proper if and only if X is -compact.F F → F F Proof. Combine the Theorem with Theorem 3.3(1). Exercises. 1. Show that a subobject of an -separated object with -compact domain is -closed. 2. Show that the assumption ofF -separation for Y is essentialF in the Theorem.F F 5.2. Perfect maps. Theorem 5.1 leads to a strengthening of Proposition 3.2(3), as follows:

Proposition. If g f ∗ with g 0, then f ∗. · ∈ F ∈ F ∈ F Proof. With f : X Y , g : Y Z, if g f ∗ and g 0, then f :(X, g f) (Y, g) is a morphism→ in /Z with→ -compact· ∈ F domain and∈ F -separated codomain· → (see Prop. 3.5 and Prop.X 4.5). Hence,F f is -proper in /ZFand also in . F X X We call a morphism -perfect if it is both -proper and -separated. With Propositions 3.2, 3.7, 4.2F and 4.4 we then obtain:F F Theorem. The class of -perfect morphisms contains all -closed subobjects, is closed under compositionF and stable under pullback. Furthermore,F if a composite morphism g f is -perfect, then f is -perfect whenever g is -separated, and g is -perfect· wheneverF f is -perfect andF stably in . Finally,F if the category F F E X 130 III. A Functional Approach to General Topology

has ﬁnite coproducts, is extensive, and if the coproduct of two -closed morphisms is -closed, also the coproduct of two -perfect morphisms isF -perfect. 2 F F F

Example. In opopen a map f : X Y is -perfect if and only if f is open and locally injectiveT (see Exercise 1→ of 4.1).F This means precisely that f is a local homeomorphism, so that every point in X has an open neighbourhood U such that the restriction U f[U] of f is a homeomorphism. In particular, local homeomorphisms enjoy all stability→ properties described by the Theorem.

5.3. Compact Hausdorff objects. Let -CompHaus denote the full subcategory of containing the objects that are bothF -compact and -Hausdorff. X F F Theorem. -CompHaus is closed under finite limits in and under -closed subobjects.F If is stable under pullback, then the ( , )-factorizationX systemF of restricts toE an ( , )-factorization system of -ECompHausM . If the category hasX finite coproducts,E M is extensive, and if the coproductF of two -closed morphismsX is -closed, then -CompHaus is closed under finite coproductsF in precisely whenF 1 + 1 -CompHausF . X ∈ F Proof. From Theorem 3.3(1) and item (vii) of Theorem 4.3 one sees that - CompHaus is closed under equalizers in ; likewise for closed subobjects. ForF closure under finite products, use TheoremX 3.3(3) and Corollary 4.3. In the ( , )- E M factorization (1) of f, Z is -compact by Theorem 3.3(2) in case = ∗, and -separated by Corollary 4.3.F For the last statement of the TheoremE oneE uses PropositionsF 3.7 and 4.4, but in order to do so we have to make sure that the morphisms 0 X and X + X X are -closed if 1 + 1 -CompHaus. For X + X →X, this follows directly→ fromF 1 + 1 -Comp∈ (see F the proof of Prop. 3.7), and→ for 0 X observe that this is the∈ Fequalizer of the injections → X / X + X in the extensive category . Hence, -closedness follows again / X F with (vii) of Theorem 4.3.

Exercises. 1. Using previous exercises, check the correctness of the given characterization of X -CompHaus for each of the following categories: op: compact Hausdorff, oc∈: F T L compact Hausdorff, opopen: discrete, opclopen: finite discrete, opZariski: T0 plus the property of ExerciseT 2 of 3.5. T T 2. Find sufficient conditions for -CompHaus to be closed under finite coproducts. F 5.4. Non-extendability of proper maps. An important property of -proper maps is described by: F Proposition. An -proper map f : M Y in cannot be extended along an -dense subobjectFm : M X with X →-HausdorffX unless m is an isomorphism. F → F Proof. Suppose we had a factorization f = g m with g : X Y , X -Hausdorff and m in -dense. Then m :(M, f) ·(X, g) is an →-dense embeddingF in /Y , withMF-compact domain and -separated→ codomain,F by (ii) of Theorem 4.3 X F F 6. Tychonoff spaces, absolutely closed spaces, compactification 131

and Propositions 3.5, 4.5. Hence, an application of Proposition 5.1 (to /Y in lieu of ) gives that m is -closed, in fact an isomorphism by CorollaryX 2.2(2). X F We shall see in 6.6 below under which circumstances the property described by the Proposition turns out to be characteristic for -properness. F 6. Tychonoﬀ spaces, absolutely closed spaces, compactiﬁcation

6.1. Embeddability and absolute closedness. In our standard setting of 2.1 we consider two important subcategories of -Haus, both containing -CompHaus. An object X of is called F F X - -Tychonoff if it is embeddable into an -compact -Hausdorff object, soF that there is m : X K in with KF -CompHaus;F - absolutely -closed if it→ is -separatedM and ∈-closed F in every -separated extension object,F so thatF every m : X KF in with K F -Haus is -closed. → M ∈ F F Denoting the respective full subcategories of by -Tych and -AC, with Proposition 5.1, Theorem 3.3(3), Corollary 4.3 weX obtain:F F Proposition. -CompHaus = -Tych -AC. F F ∩ F Examples. (1) In op, the -Tychonoff spaces are precisely the completely regular HausdorffT spacesF X, characterized by the property that for every closed set A in X and x X A, there is a continuous mapping g : X [0, 1] into the unit interval∈ with\ g[A] 1 and g(x) = 0. Using the→ Stone- ⊆ { } Cechˇ compactification βX : X βX one can prove this assertion quite easily. → (2) In oc, the -Tychonoff locales are exactly the completely regular locales (seeL II.6 andF [29], IV.1.7). (3) In op, the absolutely -closed spaces are (by definition) the so-called H-closedT spaces: see [20],F p.223. (4) In Ω- et (see 2.7) the absolutely -closed objects have been characterized in [15]S as the so-called algebraic ΩF-sets. In the case that K is given by the ground field in the category of commutative K-algebras, all such objects are subobjects of KI for some set I and are called K-algebraic . Here the K-algebraic set KI is equipped with the K-algebra of polynomial functions on KI , and its K-algebraic subsets are precisely the zero sets of sets of polynomials in K[Xi]i I . ∈ Exercises.

1. Prove that in opopen one has -Haus = -CompHaus = -Tych = -AC, given by the subcategoryT of discrete spaces.F ProveF a similar resultF for bGrp.F A 132 III. A Functional Approach to General Topology

2. Prove that in opZariski -Haus is the category of T 0-spaces, whereas -AC is the category of soberT T 0-spacesF (see Chapter II). F 3. Find examples in op of a Tychonoff space which is not absolutely closed, and of an absolutely closedT space which is not Tychonoff. (Following standard praxis we left off the prefix here.) F ext 4. Consider the category hu2 of Exercise 3 of 2.7. For an object (X, τ) of this category, let N : X 2X , x C U τ x U the “neighbourhood filter map”. Prove that → 7→ { ∈ext| ∈ } (X, τ) is -Hausdorff in hu2 if and only if N is injective, and absolutely -closed if and onlyF if N is bijective.C (Hint: For (X, τ) absolutely -closed, assume thatF N fails to be surjective, witnessed by α τ; then consider theF structure σ on Y = X + , given by σ = (τ α) U ⊆ U α , and verify that (Y, σ) is -Hausdorff.){∞} \ ∪ { ∪ {∞} | ∈ } F 6.2. Stability properties of Tychonoff objects. -Tych has the expected stability properties: F Proposition. -Tych is closed under finite limits and under subobjects in . F X Proof. With mi : Xi Ki in (i = 1, 2), also m1 m2 : X1 X2 K1 K2 is in (see 1.4(5)), and→ we canM use 3.3(3) to obtain closure× of ×-Tych→ under× finite products.M Closure under equalizers follows from closure underF subobjects, and the latter property is trivial. Remark. -AC generally fails to be closed under finite products or under equalizers. For example,F in op -AC fails to be closed under equalizers, although it is T F closed under arbitrary products, by a result of Chevalley and Frink [8]. In opopen -AC is still closed under finite products, but not under infinite ones: see ExerciseT 1F of 6.1.

6.3. Extending the notions to morphisms. Using comma categories, it is natural to extend the notions introduced in 6.1 from objects to morphisms of . Hence, a morphism f : X Y in is called X → X - -Tychonoff if (X, f) is an -Tychonoff object in /Y , which means thatF there is a factorization fF= p m with m andX p -perfect (see Prop. 3.5, 4.5); · ∈ M F - absolutely -closed if (X, f) is an absolutely -closed object in /Y , which meansF that f is -separated and, wheneverF there is a factorizationX f = p m with m Fand p -separated, then m is -closed. · ∈ M F F There is some interaction between the object and morphism notions, similarly to what we have seen for compactness and separation. Proposition. In each (1) and (2), the assertions (i)-(iii) are equivalent: (1) i. X -Tych; ii. every∈ F morphism f : X Y is -Tychonoff; iii. there is an -Tychonoff→ map fF: X Y with Y -CompHaus; (2) i. X -AC;F → ∈ F ii. every∈ F morphism f : X Y with Y -Haus is absolutely -closed; → ∈ F F 6. Tychonoff spaces, absolutely closed spaces, compactification 133

iii. there is an absolutely -closed morphism f : X Y with Y - CompHaus. F → ∈ F Proof. For morphisms f : X Y , m : M K in consider the diagram → → X X (23) w GG f ww GG m ww GG ww GG {ww G# Y o Y K / K p1 × p2 If K -CompHaus, then p1 is -perfect, and m = p2 < f, m > implies < f, m∈ F> ; this proves (1) i F ii. Similarly, for (2)· iii i, the∈ hypotheses M ∈ M ⇒ ⇒ m and K -Haus give < f, m > and p1 0, so that < f, m > must ∈ M ∈ F ∈ M ∈ F be -closed since f is absolutely -closed; furthermore, p2 is -closed since Y is F F F -compact, so that m = p2 < f, m > is -closed. F Let us now assume f =· p m with mF . If p : K Y is -perfect and Y -CompHaus, then also K· -CompHaus,∈ M by Theorems→ 3.3(1),F 4.3(iii). This proves∈ F (1) iii i. For (2) i ii,∈ assume F both p and Y to be -separated, so that also K is -separated;⇒ now⇒ the hypothesis X -AC givesF that m is -closed, as desired.F ∈ F F The implications ii iii are trivial in both cases: take Y = 1. ⇒ Of course, the Proposition just proved may be applied to the slices of rather than to itself and then leads to stability properties for the classes of -TychonoffX and of absolutelyX -closed maps, the proof of which we leave as exercises.F F Exercises. 1. The class of -Tychonoff maps is stable under pullback. If g f is -Tychonoff, so is f; conversely,F f -Tychonoff and g -perfect imply g f -Tychonoff.· F 2. If g f is absolutelyF -closed and g F -separated, then· f Fis absolutely -closed; conversely,· if f is absolutelyF -closedF and g -perfect, then g f is absolutelyF - closed. F F · F

Remark. Unlike spaces, -Tychonoff maps in op do not seem to allow for an easy characterization in termsF of mapping propertiesT into the unit interval. Other authors (see [16, 41, 33]) studied separated maps f : X Y with the property that for every closed set A X and every x X A there→ is an open neighbourhood ⊆ ∈ \ 1 U of f(x) in Y and a continuous map g : f − [U] [0, 1] with g(x) = 0 and 1 → g[A f − [U]] 1 . In case Y = 1 this amounts to saying that X is completely regular∩ Hausdorff,⊆ { hence} Tychonoff. For general Y , maps f with this property are -Tychonoff, i.e. restrictions of perfect maps, but the converse is generally false: seeF [51].

6.4. Compactification of objects. An -compactification of an object X is given by an -dense embedding X K withFK -CompHaus. Of course, only objects in F-Tych can have -compactifications.→ ∈ For F our purposes it is important to be providedF with a functorialF choice of a compactification for every X -Tych. ∈ F 134 III. A Functional Approach to General Topology

Hence, we call an endofunctor κ : -Tych -Tych which comes with a natural transformation F → F

κX : X κX (X -Tych) −→ ∈ F a functorial -compactification if F - each κX is an -dense embedding - each κX is -compactF and -separated. F F By illegitimally denoting the endofunctor and the natural transformation by the same letter we follow standard praxis in topology. In op, the prime example for a functorial -compactification is provided by the Stone-T Cechˇ compactification F βX : X βX. −→ These morphisms serve as reflexions, showing the reflexivity of -CompHaus in -Tych. We shall revisit this theme in Section 11 below; here weF just note that theF universal property makes the first of the two requirements for a functorial -compactification redundant, also in our general setting: F Proposition. If -CompHaus is reflective in -Tych, with reflexions βX : X βX, then theseF provide a functorial -compactification.F → F Proof. We just need to show that each βX is an -dense embedding. But for X -Tych we have some m : X K in withF K -CompHaus, which ∈ F → M ∈ F factors as m = f βX , by the universal property of βX . Hence βX . If · ∈ M βX = n g with n : N βX -closed, then N -CompHaus, and we can · → F ∈ F apply the universal property again to obtain h : βX N with h βX = g. Then → · n h βX = βX shows n h = 1βX , so that n is an isomorphism. Consequently, βX is· -dense.· · F 6.5. Compactification of morphisms. In order to take full advantage of the presence of a functorial -compactification κ, we should extend this gadget from objects to F morphisms, as follows: for f : X Y in -Tych we form the pullback Pf = Y κY → F × κX and the induced morphism κf making the following diagram commutative: X (24) 0APPP 0 A PP κ 0 AA PP X 0 A κfPP 0 A PPP 0 A PP 00 P( f 0 Pf / κX 0 mf 00 0 pf κf 00 κY Y / κY Calling a morphism f : X Y -initial if every -closed subobject of X is the inverse image of an -closed→ subobjectF of Y underFf, we observe: F Proposition.

(1) κf , and pf is -perfect, with Pf -Tych. ∈ M F ∈ F (2) If every morphism in is -initial, then κf is -dense. M F F 6. Tychonoﬀ spaces, absolutely closed spaces, compactiﬁcation 135

Proof. (1) Since κX = mf κf we have κf . As a morphism in - CompHaus, κf is -perfect (by· Prop.∈ M 5.1 and Theorem∈ M 4.3), and so is its pullbackF F pf . Since -Tych is closed under finite limits in , Pf -Tych. (2) The givenF hypothesis implies the following cancellationX ∈ F property for -dense subobjects: if n k is -dense with n, k , then k is -dense. This mayF then · F ∈ M F be applied to κX = mf κf since mf , as a pullback of κY . · ∈ M Hence, we should think of κf :(X, f) (Pf , pf ) as of an -compactification of (X, f) in -Tych/Y , especially under the→ hypothesis given inF (2). F Exercises. 1. Show P κX in case Y = 1 and κ 1. f ∼= 1 ∼= 2. Show that by putting κ(X, f) := (Pf , pf ) one obtains a functorial -compactification for -Tych/Y , under the same hypothesis as in (2) of the Proposition.F (Here we think of F-Tych/Y as a full subcategory of /Y that has inherited the factorization system andF the closed system from there.) X

6.6. Isbell-Henriksen characterization of perfect maps. In case = op the following theorem goes back to Isbell and Henriksen (see [23]). CategoricalX T versions of it can be found in [9] and [48]. Theorem. Let κ be a functorial -compactiﬁcation and let every morphism in be -initial. Then the following assertionsF are equivalent for f : X Y in -TychM : F → F (i) f is -perfect; (ii) f cannotF be extended along an -dense embedding m : X Z with Z -Hausdorﬀ unless m is an isomorphism;F → (iii) theF naturality diagram κ X X / κX (25)

f κf

κY Y / κY is a pullback diagram. Proof. (i) (ii) was shown in 5.4. ⇒ (ii) (iii): By Prop. 6.5, κf of (24) is an -dense embedding, hence an isomorphism⇒ by hypothesis. Consequently, (25) coincidesF with the pullback square of (24), up to isomorphism. (iii) (i): By hypothesis f = pf , hence f is -perfect by 6.5. ⇒ ∼ F Example. For = op and κ = β the Stone-Cechˇ compactification, by item (iii) of the Theorem perfectX T maps f : X Y between Tychonoff spaces are characterized by the property that their extension→ βf : βX βY maps the remainder βX X into the remainder βY Y . → \ \ 6.7. Antiperfect-perfect factorization. Suppose that, in our general setting of 2.1, κ = β is given by reflexivity of -CompHaus in -Tych. Then there is another F F 136 III. A Functional Approach to General Topology

way of thinking of the factorization f = pf βf established in diagram (24). With denoting the class of -perfect morphisms· in -Tych, and with PF F F := m : X Y m ,X,Y -Tych, βm : βX βY iso , AF { → | ∈ M ∈ F → } whose morphisms are also called -antiperfect, one obtains: F Theorem. ( , ) is a (generally non-proper) factorization system of -Tych, provided thatAF -PCompHausF is reﬂective in -Tych and every morphismF in is -initial. F F M F Proof. The outer square and the upper triangle of

βX X / βX (26) {= {{ βf { βf {{mf {{ Pf / βPf βPf

are commutative, with βf := β(βf ). According to Theorem 2.7 of [27], the assertion of the Theorem follows entirely from the universal property of β once we have established commutativity of the lower triangle of (26). For that, let u : E Pf → be the equalizer of βP , βf mf , which is -closed since βPf is -separated (see f · F F Theorem 4.3). Since βf factors through u by a unique morphism l : X E, and → since βf is -dense by Prop. 6.5, u is an isomorphism, and we have βf mf = βP . F · f An important consequence of the Theorem is that essentially it allows us to carry the hypothesis of reﬂectivity of -CompHaus in -Tych from to its slices, as follows. Consider again a morphismF f : X Y withF X,Y X-Tych. Then Prop. 6.5(1) shows that the object (X, f) in →/Y (in fact: ∈-Tych F /Y ) is - Tychonoﬀ. Now, with the Theorem, one easily shows:X F F

Corollary. Under the provisions of the Theorem, βf :(X, f) (Pf , pf ) is a re- flexion of the -Tychonoff object (X, f) of -Tych/Y into the→ full subcategory of -compact HausdorffF objects of /Y . F 2 F X Hence, the construction given by diagram (24) with κ = β is really the “Stone- Cechˇ -compactification” of -Tychonoff objects in /Y , provided that we re- strictF ourselves to those objectsF (X, f : X Y ) for whichX X,Y are -Tychonoff objects in . → F X Exercise. Work out the details of the proof of the Theorem and the Corollary.

7. Open maps, open subspaces

7.1. Open maps. In the standard setting of 2.1, a morphism f : X Y is said to 1 → reﬂect -density if f − [ ] maps -dense subobjects of Y to -dense subobjects F − F F of X. The morphism f is -open if every pullback f 0 of f (see diagram (11)) F 7. Open maps, open subspaces 137

reflects -density. A subobject is -open if its representing morphism is -open. By definitionF one has F F + := f f -open = f f reflects -density ∗ F { | F } { | F } in the notation of Section 3. Since reflection of -density is obviously closed under composition, one obtains immediately some stabilityF properties for -open maps, just as for -proper maps. F F Proposition. (1) The class + contains all isomorphisms, is closed under composition and stable underF pullback. (2) If g f + with g monic, then f +. · ∈ F + ∈ F+ (3) If g f with f ∗, then g . · ∈ F ∈ E ∈ F

Proof. For (1), (2), proceed as in Prop. 3.2. (3): A pullback of g f with f ∗ · ∈ E has the form g0 f 0 with f 0 a pullback of f and g0 a pullback of g. So, it suffices · ∈ E to check that g reflects -density if g f does and f ∗. But for every -dense subobject d of the codomainF of g one obtains· from Prop.∈ E 1.8 that F 1 1 1 1 g− [d] = f[f − [g− [d]]] = f[(g f)− [d]] · is an image of an -dense subobject under f. But since f , trivially f[ ] preserves -densityF (by Cor. 2.2(1),(3)). ∈ E − F Analogously to Prop. 2.1 we can state: Corollary. (1) The -open subobjects of an object X form a (possibly large) subsemilat- tice ofF the meet-semilattice subX. 1 (2) For every morphism f, f − [ ] preserves -openness of subobjects. (3) For every -open morphism−f, f[ ] preservesF -openness of subobjects provided thatF is stable under pullback.− F E Proof. (1) follows from the Proposition, and for (2) remember that -openness of subobjects is, by definition, pullback stable. (3) If f and m areF both open, ∈ M then -openness of f m = f[m] e with e ∗ yields -openness of f[m], by the F · · ∈ E F Proposition.

7.2. Open maps of topological spaces. It is time for a “reality check” in terms of our standard examples. Proposition. In op (with = closed maps ), the following assertions are equivalent for a continuousT mapFf : X{ Y : } → (i) f is -open; F 1 1 (ii) for all subspaces N of Y , f − [N] = f − [N]; (iii) f is open, i.e. f[ ] preserves openness of subspaces. − 138 III. A Functional Approach to General Topology

1 Proof. (i) (ii): Given N Y , by hypothesis, the restriction f 0 : f − [N] N ⇒ ⊆ 1 1 → of f reflects the density of N in N, hence f − [N] = f − [N]. 1 1 (ii) (iii): Given O X open, f − [Y f[O]] = f [Y f[O]] X O = X O, ⇒ ⊆ \ − \ ⊆ \ \ hence Y f[O] Y f[O], that is f[O] is open. (iii) (i):\ Openness⊆ \ of maps (in the sense of preservation of openness of subspaces) is easily⇒ shown to be stable under pullback. Hence, it suffices to show that an open map f : X Y reflects density of subspaces. But if D Y is dense, then 1 → ⊆ f[X f − [D]] is open, by hypothesis, and this set would have to meet D under \ 1 1 the assumption that X f [D] is not empty, which is impossible. Hence, f − [D] \ − is dense in X. We emphasize that a map in op which reflects -density need not be open (see Exercise 1 below); hence, the stabilityT requirementF in Definition 7.1 of -openness is essential. F Exercises. 1. Show that in op the embedding of the closed unit interval into R reflects density. T 2. In opopen, D X is -dense if and only if D meets every non-empty closed set in X.T Hence, if X⊆is a T 1-spaceF (so that all singleton sets are closed), only X itself is a dense subobject of X. Show that every proper (=stably-closed) map is -open, but not conversely. F 3. In opclopen, D X is -dense if and only if D meets every non-empty clopen set in XT . Prove that⊆f is F-open if and only if f is -proper (i.e., every pullback of f maps clopen subsets ontoF clopen subsets). F 4. Show that in a topos with a universal closure operator (see 2.5), every morphism is -open. F 7.3. Open maps of locales. In oc, -open morphisms f : X Y are characterized L F 1 → like in op as those that have stably the property that f − [ ] commutes with the usual closure,T just as in (ii) of Prop. 7.2; this follows formally− from the fact that the closure in oc is given by an idempotent and hereditary closure operator (see [18]). But moreL importantly we need to compare this notion with the usual notion of open map of locales (see Chapter II): Proposition. If f[ ] maps open sublocales to open sublocales (i.e., if f is open), then f is -open− in oc. F L Proof. Since openness of localic maps is stable under pullback (see Theorem II 5.2), it suffices to show

1 1 f − [n] = f − [n] for f : X Y open and any sublocale n : N Y . For this it suﬃces to show → 1 f ∗[c(n)] = c(f − [n]), where c(n) = b OY n∗(b) = 0 (see II, 2.9 and 3.6), which means: W{ ∈ | } f ∗(b) b OY, n∗(b) = 0 = a OX m∗(a) = 0 , _{ | ∈ } _{ ∈ | } 7. Open maps, open subspaces 139

1 with m = f − [n]: M X. Now, for this last identity, “ ” is trivial, while “ ” ≤ ≥ follows when we put b = f!(a) for every a with m∗(a) = 0 and use

n∗(f!(a))) = (f 0)!(m∗(a)), where f! is the left adjoint of f ∗ and f 0 : M N is the restriction of f. → After the authors posed the converse statement of the Proposition as an open problem, in April 2002 P.T. Johnstone proved its validity:

Theorem. The -open maps in oc are precisely the (usual) open maps of locales. F L

For its rather intricate proof we must refer the Reader to [31]. The paper also exhibits various subtypes of openness. For example, like in op, also in oc there are examples of morphisms reﬂecting -density which failT to be -open,L as also shown in II.5.2. F F

7.4. Local homeomorphisms. Proposition 7.1 shows that, if is stable under pullback in , then + is a new ( , )-closed class in (seeE 2.1), and in view of Exercise 1X of 4.3 andF Example 5.2,E M it would make senseX to deﬁne: X -discrete : X +-separated F ⇔ F δX : X X X -open ⇔ → × F f : X Y local -homeomorphism : X +-perfect → F ⇔ F f and δf : X X Y X -open ⇔ → × F Here we cannot explore these notions further, but must leave the Reader with:

Exercises. 1. Collect stability properties for -discrete objects and local -homeomorphisms, from the properties already shown forF -separated objects and F -proper morphisms. 2. In a topos with a universal closureF operator (see 2.5) everyF object is -discrete and every morphismS is a local -homeomorphism. F F

7.5. Sums of open maps. The proof of the following Proposition is left as an exercise as well:

Proposition. Assume that the finitely-complete category has finite coproducts X and is extensive, and that the classes and 0 are closed under finite coproducts. Then: M F

f1 + f2 -dense f1, f2 -dense, F ⇔ F f1 + f2 -open f1, f2 -open. F ⇔ F 2 140 III. A Functional Approach to General Topology

8. Locally perfect maps, locally compact Hausdorﬀ spaces

8.1. Locally perfect maps. In the setting of 2.1, a morphism f : X Y is locally -perfect if it is a restriction of an -perfect morphism p : K Y →to an -open Fsubobject u : X K: F → F → K (27) u > A p || AA || A X / Y f

Such morphisms are in particular -Tychonoff. An object X is locally -compact Hausdorff if X 1 is locally -perfect,F that is: if there is an -open embeddingF u : X K with→K -CompHaus.F If it is clear that X is -Hausdorff,F we may simply→ call X locally∈ F-compact. F F Examples. (1) In op, X is locally -compact Hausdorff if and only if X is Haus- dorffT and locally compact,F in the sense that every point in X has a base of compact neighbourhoods. By constructing the Alexandroff one- point compactification for such spaces, one sees that they are locally -compact. Conversely, compact Hausdorff spaces are locally compact, andF local compactness is open-hereditary. (2) Every locally compact Hausdorff locale (in the sense of Chapter II, 7) is an open sublocale of its Stone-Cechˇ compactification (see II.6.7), hence the embedding is -open by Prop. 7.3 and X is a locally -compact Hausdorff object inF oc. We conjecture that also the converse propositionF is true. L

8.2. First stability properties. The following properties are easy to prove:

Proposition. (1) Every morphism representing an -closed or an -open subobject is a locally -perfect morphism, and soF is every -perfectF morphism. (2) The classF of locally -perfect morphisms is stableF under pullback; moreover, if in the pullbackF diagram (11) both f and g are locally -perfect, F so is g f 0 = f g0. · · Proof. (1) -closed subobjects give -perfect morphisms. Hence, the assertion follows by choosingF u or p in (27) to beF an identity morphism. (2) Both -open subobjects and -perfect morphisms are stable under pullback. Furthermore,F if f = p u, g = q v Fare both locally -perfect, the pullback diagram · · F 8. Locally perfect maps, locally compact Hausdorﬀ spaces 141

(11) is decomposed into four pullback diagrams, as follows:

u p0 0 / / (28) · · · v0 v00 v u00 p00 / / · · · q0 q00 q u p / / · · ·

Hence g f = (q v) (p0 u0) = (q p00) (v00 u0), with q p00 -perfect and v00 u0 an -open· subobject.· · · · · · · F · F

Corollary. The full subcategory -LCompHaus of locally -compact Hausdorff objects in is closed under finiteF products and under -openF subobjects in . X F X Proof. For the first statement, choose Y = 1 in diagram (11) and apply assertion (2) of the Proposition. The second statement is trivial.

Exercise. Prove: if U X in subX is -open and A X in subX -closed, then → F → F U A X is locally -perfect. ∧ → F

8.3. Local compactness via Stone-Cech.ˇ For the remainder of this Section, we assume that -CompHaus is reﬂective in -Tych, so that in particular we have a functorial -compactiﬁcationF F F

βX : X βX (X -Tych) −→ ∈ F at our disposal. We shall also use its extension to morphisms, as described in diagram (24), with κ = β.

Theorem (Clementino-Tholen [13]). An object X is locally -compact Hausdorff F if and only if X is -Tychonoff and βX is -open. F F Proof. Sufficiency of the condition is trivial. For its necessity, let u : X K be an -open subobject with K -CompHaus. With the unique morphism→ F ∈ F f : βX K with f βX = u we can then form the pullback diagram → · u P 0 / βX (29)

f 0 f u X / K 142 III. A Functional Approach to General Topology

and have an induced morphism d : X P with f 0 d = 1X and u0 d = βX. Since → · · u0 is -open and βX -dense (by Cor. 7.1(2) and Prop. 6.4), the pullback diagram F F 1 X X / X (30)

d βX

u0 P / βX

shows that d is -dense. On the other hand, the equalizer diagram F

d 1P X / P // P (31) d f 0 · shows that d is -closed (since P -Haus, by Cor. 4.3). Therefore, d is an F ∈ F isomorphism (see Cor. 2.2(2)), and βX = u0 is -open. ∼ F With Corollary 6.7 we obtain immediately from the Theorem:

Corollary. If every morphism in is -initial, then a morphism f : X Y M F → with X,Y -Tych is locally -perfect if and only if the morphism βf : X Pf making diagram∈ F (24) (with β =Fκ) commutative is -open. → 2 F

8.4. Composites of locally perfect maps. We can now embark on improving some of the properties given in 8.2.

Proposition. Let every morphism in be -initial. Then the class of locally - perfect morphisms with -TychonoﬀM domainF and codomain is closed under com-F position. F

Proof. It obviously suﬃces to show that any composite morphism h = v p with p : K Y -perfect and v : Y L an -open subobject is locally -perfect.· → F → F F For that we must show that its -antiperfect factor βh : K Ph is -open. But the β-naturality diagram for h decomposesF as → F

βK K / βK (32)

p 1 βp

βY Y / βY

v βv

βL L / βL 8. Locally perfect maps, locally compact Hausdorﬀ spaces 143

with the upper square a pullback, since p is -perfect (see Theorem 6.6). Now (32) decomposes further, as F

βh mh K / Ph / βK (33)

p 2 3 βp

βv mv Y / Pv / βY

v pv 4 βv

1L βL L / L / βL Here 3 is a pullback diagram, since 4 and the concatenation 3 & 4 are pullback diagrams. Now, with 1 = 2 & 3 also 2 is a pullback diagram. Since the - F antiperfect factor βv of the -open (and therefore locally -perfect) morphism v F F is -open, also its pullback βh must be -open. F F Corollary. If every morphism in is -initial, then the following assertions are equivalent for X -Tych: M F ∈ F (i) X is locally -compact; (ii) every morphismF f : X Y with Y -Tych is locally -perfect; (iii) there is a locally -perfect→ map f : X∈ F Y with Y -FLCompHaus. F → ∈ F Proof. (i) (ii): In diagram (24) with κ = β, βf is open since βX is, and since ⇒ mf is monic, by Prop. 7.1(2). (ii) (iii) is trivial. (iii) (ii): !X =!Y f is locally -perfect, by the Proposition. ⇒ ⇒ · F 8.5. Further stability properties. We can now improve the assertions made in 8.2: Theorem. If every morphism in is -initial, then the full subcategory - LCompHaus of locally -compactM HausdorffF objects is closed under finite limitsF and under -closed or F-open subobjects in . F F X Proof. We already stated closure under finite products and -open subobjects in Cor. 8.2. Closure under -closed subobjects follows from Prop.F 8.4, and this fact F then implies closure under equalizers, by Thm. 4.3(viii). 8.6. Invariance theorem for local compactness. We finally turn to invariance of local -compactness under -perfect maps. For that purpose let us call a mor- phismFf : X Y -final ifF a subobject of Y is -closed whenever its inverse image under f→is -closed.F Clearly, with Prop. 1.8(2)F one obtains: F Lemma. If every pullback of f along a morphism in lies in , then f is -final. ∈ F M E F Translated into standard topological terms, the Lemma asserts that in op closed surjective maps are quotient maps. Now, in op, to say that the mapT T 144 III. A Functional Approach to General Topology

1 f : X Y has the property that N Y is closed as soon as f − [N] X is → ⊆ 1 ⊆ closed is trivially equivalent to saying that N is open whenever f − [N] is open. In general, however, we may not assume such equivalence.

Example. In opopen, -ﬁnality takes on the usual meaning: N Y is open (= T F 1 ⊆ -closed) as soon as f − [N] is open in X. However, every map f : X Y in F 1 → opopen has the property that N Y is -open as soon as f − [N] is -open, providedT that X and Y are T 1-spaces.⊆ Indeed,F in this case all subobjectsF are - open, because there are only isomorphic -dense subobjects: see Exercise 2 of 7.2.F F

Theorem. Let every morphism in be -initial, let be stable under pullback along -morphisms and let every M-finalF morphism inE have the property that a subobjectM of its codomain is -openF whenever its inverseE image is -open. Then, for every -perfect morphismF f : X Y with X,Y -Tych, oneF has: F → ∈ F (1) if Y is locally -compact, so is X; (2) if f and XFis locally -compact, so is Y . ∈ E F Proof. (1) is a special case of Cor. 8.4 (iii) (i). For (2), first observe that - ⇒ F density of f and βY gives the same first for βf βX = βY f and then for βf. But βf : βX βY is also -closed (see Prop. 5.1),· hence βf· (see Exercise 1 of 2.2). Furthermore,→ by theF Lemma, βf is -final. Since the ∈-perfect E morphism f makes F F

βX X / βX (34)

f βf

βY Y / βY a pullback diagram with 1 βX = (βf)− [βY ]

-open, by hypothesis also βY must be -open. F F Remark. Analyzing the condition of pullback stability of along embeddings one observes that this stability property is needed only alongE -closed embeddings. F Exercise. Using Prop. 7.5, give suﬃcient conditions for -LCompHaus to be closed under F ﬁnite coproducts in . X

9. Pullback stability of quotient maps, Whitehead’s Theorem

9.1. Quotient maps. The formation of quotient spaces is an important tool in topology when constructing new spaces from old: given a space X and a surjective mapping f : X Y , one provides the set Y with a topology such that any B Y → 1 ⊆ is open (closed) if f − [B] X is open (closed). In our general setting of 2.1, we ⊆ 9. Pullback stability of quotient maps, Whitehead’s Theorem 145

have called a morphism f : X Y in -final (see 8.6) if any b subY is - 1 → XF ∈ F closed whenever f − [b] subX is -closed, and we refer to an -final morphism in as an -quotient map∈ . CertainF pullbacks of -final morphismsF are -final: E F F F 1 Proposition. The restriction f 0 : f − [B] B of an -final morphism f : X Y to an -closed subobject b : B Y is -final.→ F → F → F Proof. For a subobject c : C B one has → 1 1 1 f − [b c] = f − [b] (f 0)− [c]. · ∼ · 1 1 Hence, if (f 0)− [c] is -closed, so is f − [b c] and also b c, by hypothesis. But -closedness of b c impliesF the same for c,· by Prop. 3.2(3).· F · In general, -finality fails badly to be stable under pullback, already for = op and in veryF elementary situations. X T Example. We consider the quotient map f : R R/Z, so that f(x) = f(y) if and Z R →1 N R only if x y , and the subspace S := n n , n 2 of . Then the map − ∈ \{ | ∈ ≥ }

g := idS f : S R S R/Z × × → × is the pullback of f along the projection S R/Z R/Z, but fails to be a quotient map. Indeed, the set × → 1 π i + 1 B := + , ) i, j 2 { i j j | ≥ } 1 is closed in S R and g− [g[B]] = B, but g[B] is not closed in S R/Z (the point × × (0, f(1)) lies in g[B] g[B]). Let us also note\ that the map f is closed; hence: the product of two closed quotient maps needs neither to be closed nor a quotient map.

Exercise. Verify the claims made in the Example and show that the space S fails to be locally compact.

9.2. Beck-Chevalley Property. In what follows we would like to extend Propo- sition 9.1 greatly by showing that -quotient maps are stable under pullback along -perfect maps. But this needsF some preparations and extra conditions. The conditionF given in the following proposition is known as (an instance of) the Beck-Chevalley Property. Proposition. The class of the factorization system ( , ) of is stable under pullback if and only if, forE every pullback diagram E M X g U / V (35)

q p X / Y f

1 1 and every a subX, g[q− [a]] = p− [f[a]]. ∈ ∼ 146 III. A Functional Approach to General Topology

Proof. Considering a ∼= 1X one sees that the condition is suﬃcient for pullback stability: with Exercise 2 of 1.6, f implies g . For its necessity we consider the commutative diagram ∈ E ∈ E g 1 U 1 / V (36) q− [a] ; p− [f[a]] 9 vv ss vv q sss v g0 s 1 1 p q− [A] / p− [f[A]]

f q0 X / Y a : s9 vv p0 s vv sss vv s f[a] A / f[A] f 0

It suffices to show that the morphism g0 (induced by the pullback property of 1 p− [f[A]]) is in . But since all other vertical faces are pullback diagrams, also E the front face of (36) is a pullback diagram; consequently, with f 0 we obtain ∈ E q0 , by hypothesis on . ∈ E E 9.3. Fibre-determined categories. We showed in 3.5 that in op proper maps are characterized as closed maps with compact fibres, but thatT this characterization fails in general. Since in what follows we would like to make crucial use of it, we say that our category is fibre-determined (with respect to its ( , )-closed system ) if the followingX two conditions are satisfied: E M F 1. a morphism f lies in if and only if all of its fibres have points, that is: if for every fibre F ofEf there is a morphism 1 F ; → 2. a morphism f is in ∗ if and only if f[ ] preserves -closedness of subobjects and f hasF -compact fibres. − F F Of course, in condition 2 “only if” comes for free (see Prop. 2.1(3) and Cor. 3.5). Condition 1 means equivalently that contains exactly those morphisms with respect to which the terminal object 1E is projective, as the Reader will readily check: Exercises. 1. Prove that for a morphism f : X Y all fibres have points if and only if the hom-map (1, f): (1,X) (1,Y→) is surjective. 2. Prove thatX the classX of all morphisms→ X f for which (1, f) is surjective is stable under pullback in . X X As a consequence we see that condition 1 forces to be stable under pullback. We note that, granted pullback stability of , in theE remainder of this section we shall use only the “only if” part of conditionE 1. Observe that op is fibre-determined (by Theorem 3.5), and so is opopen (trivially, see ExerciseT 1 of 3.5), but that op/B is not (by Example 3.5).T T 9.4. Pullbacks of quotient maps. Next we establish a result which, even in the case = op, was established only recently (see [10]): X T 9. Pullback stability of quotient maps, Whitehead’s Theorem 147

Theorem (Richter-Tholen [46]). If is fibre-determined, then every pullback of an -quotient map along an -perfectX map is an -quotient map. F F F Proof. We consider the pullback diagram (35) with an -perfect map f and an - quotient map p. In order to show that q is an -quotientF map as well, for a subFX 1 F ∈ we assume q− [a] -closed and must show that a is -closed. First, with Prop.F 9.2 and the Exercises 9.3 we obtainF 1 1 g[q− [a]] ∼= p− [f[a]], and this subobject is -closed since g (as a pullback of f ∗) is -closed. Whence f[a] is -closed,F by hypothesis on p. Therefore it suffices∈ F to showF that the F morphism f 0 of diagram (36) is -proper, since then -properness of F F f[a] f 0 = f a · · and -separatedness of f give -closedness of a, with Prop. 5.2. F F Now, in order to show -properness of f 0 we invoke condition 2 of 9.3 and first F show that f 0[ ] preserves -closedness of subobjects. Hence we consider b subA and note − F ∈ 1 1 g0[(q0)− [b]] ∼= (p0)− [f 0[b]] with the Beck-Chevalley Property again, applied to the front face of (36). This subobject is -closed since the morphism g0 is -closed; in fact, since F F 1 1 g q− [a] = p− [f[a]] g0 · · is -proper, so is g0, by Prop. 3.2(3). Furthermore, p0 is an -quotient map, by F F Prop. 9.1, and we can conclude that f 0[b] is -closed. F Lastly, it remains to be shown that f 0 has -compact fibres. But by condition F 1 and Exercises 9.3, any point z : 1 f[A] factors as z = p0 w, with w : 1 1 → · → p− [f[A]]. With F , G denoting the fibres of f 0, g0 belonging to z, w, respectively, we can form the commutative diagram

g 1 0 1 q− [A] / p− [f[A]] (37) w; t: ww q0 ttw ww ttt G / 1 p0

f 0 q00 f[A] ; A / vv t9 vv ttz vv tt F / 1 t Its front face is a pullback diagram since the back-, top- and bottom faces are pullback diagrams. Hence, q00 is an isomorphism, and F = G is -compact since ∼ F g0 is -proper, as shown earlier. F Corollary. For ﬁbre-determined and K an -compact -Hausdorﬀ object, with X F F p : V Y also 1K p : K V K Y is an -quotient map. → × × → × F 148 III. A Functional Approach to General Topology

Proof. Apply the Theorem to the -perfect projection K Y Y . F × → Exercises.

1. By applying the Theorem to opopen, conclude that pullbacks of quotient maps along local homeomorphisms are quotientT maps. 2. Verify that when is stable under pullback, the class + of -open maps in is ( , )-closed. WhatE does the Theorem say when we exchangeF F for +? X E M F F

9.5. Whitehead’s Theorem. An +-final morphism f : X Y has, by definition, F 1 → the property that b subY is -open whenever f − [b] subX is -open. From Prop. 9.1 applied to∈ + in lieuF of one obtains immediately:∈ F F F 1 + Lemma. The restriction f 0 : f − [B] B of an -final morphism f : X Y to an -open subobject b : B Y is →+-final. F → 2 F → F In op, +-quotient maps (= +-final maps in ) coincide with -quotient maps.T If thisF happens in , we canF use the lemma inE order to upgradeF Theorem 9.4 to: X Theorem. If is fibre-determined and if +-quotient maps coincide with - quotient maps,X then every pullback of an -quotientF map along a locally -perfectF map is an -quotient map. F F 2 F Corollary. If is fibre-determined and if +-quotient maps coincide with - quotient maps,X then for every locally -compactF -Hausdorff object K and everyF F F -quotient map p, also 1K p is an -quotient map. 2 F × F In op, the assertion of the Corollary is known as Whitehead’s Theorem. T

10. Exponentiable maps, exponentiable spaces

10.1. Reflection of quotient maps. In the previous section we have seen that, under certain hypotheses, locally -perfect maps have (stably) the property that -quotients pull back along them.F This is a fundamental property which is worth investigatingF separately. Analogously to the terminology employed in the definition of -open maps, we say that a morphism reflects -quotients if in every pullback diagramF (35) with p also q is an -quotient map, andF f is -exponentiable if every F F pullback f 0 of f (see diagram (11)) reflects -quotients. Hence, F exp := f f -exponentiable = f f reflects -quotients ∗ F { | F } { | F } in the notation of Section 3. The reason for this terminology will become clearer at the end of 10.2 below, and fully transparent in 10.9. In this terminology, since (locally) -perfect maps are pullback stable, we proved in Theorems 9.4 and 9.5: F 10. Exponentiable maps, exponentiable spaces 149

Corollary. Let be ﬁbre-determined. Then every -perfect map is -exponentiable. Even everyX locally -perfect map has this property,F if the notionsF of +- quotient map and -quotientF map are equivalent in . F 2 F X

Exercise. Conclude that in op both perfect maps and local homeomorphisms are - T F exponentiable.

Example. For a space X, the map X 1 in op always reflects -quotients (since the pullback of V 1 along it is the→ projectionT X V X),F but generally not every pullback of X→ 1 reflects -quotients: see× Example→ 9.1. Hence, in op, reflection of -quotients→ is a propertyF properly weaker than -exponentiability.T F F Before developing a theory of -exponentiability, we should provide a first justification for the terminology. F 10.2. Exponentiable maps. For every morphism f : X Y we have the pullback functor →

f ∗ : /Y /X X → X which sends an object (V, p) in /Y to (U, q) = (X Y V, proj ) in /X (see X × 1 X diagram (35)), and for a morphism h :(V, p) (V 0, p0), f ∗(h):(U, q) (U 0, q0) → → has underlying -morphism 1X h : X Y V X Y V 0. Of course, the functor 1 X × × → × f − [ ] of 1.7 is just a restriction of f ∗. − g0 U 0 / V 0 (38) f ∗(h) = h > || || || g || U / V q0 p0 q p Ø f Ù X / Y

Proposition. f is -exponentiable if and only if the functor f ∗ maps an -quotient morphism of /YF to an -quotient morphism of /X. F X F X Proof. If f is -exponentiable and h :(V, p) (V 0, p0) an -quotient map, then F → F g0 (as a pullback of f) reﬂects the -quotient map h in , hence its pullback f ∗(h) F X along g0 is an -quotient map in and also in /X. Conversely, considering any F X X pullback g0 of f and an -quotient map h in , we can consider this as a morphism in /Y and argue in theF same way as before.X X Exercises. 1. Prove that if has a certain type of colimits, so does /B, and the forgetful functor X X ΣB : /B preserves them. X → X 2. For = op, prove that f ∗ preserves coproducts for every f. X T 3. For = op, prove that f ∗ preserves coequalizers if and only if f ∗ preserves - quotientX morphisms.T F 150 III. A Functional Approach to General Topology

4. Conclude that in op (as well as in opopen) the -exponentiable morphisms f are T T F exactly those for which f ∗ preserves all small colimits. 5. Apply Freyd’s Special Adjoint Functor Theorem (see Mac Lane [35]) to prove that for an -exponentiable morphism f in op the functor f ∗ actually has a right adjoint. F T Exercise 5 identiﬁes the -exponentiable morphisms f in op (and in opopen) F T T as those for which f ∗ has a right adjoint functor, a property that may be considered in any category with pullbacks and that is equivalent to the absolute (=“ - free”) categoricalX notion of exponentiability; see 10.8 below. F 10.3. Stability properties. We shall collect some properties of the class exp of -exponentiable morphisms. First we note: F F Lemma. If g f is -ﬁnal, so is g. · F 1 1 1 Proof. If for a subobject c of the codomain of g g− [c] is -closed, so is f − [g− [c]] 1 F = (g f)− [c]. Hence -closedness of c follows with the hypothesis. · F Following the terminology used in topology we call a morphism f in an - biquotient map if every pullback of f is an -quotient map. In op other namesX F in use are universal quotient map and descentF map. In order notT to divert too much, we leave it to the Reader to establish their characterization in terms of points and open sets. Exercises.

1. Show that every -proper map in ∗ is an -biquotient map. 2. Conclude that if theF notions of +-quotientE F map and -quotient map are equivalent F F in , then every -open map in ∗ is an -biquotient map. 3. * ShowX that in opF f : X Y is anE ( -)biquotientF map if and only if, for every point T → F 1 y Y and every open cover Ui i I of the fibre f − y, the system f[Ui] i F covers∈ some neighbourhood of{ y |in∈Y ,} for some finite F I. (See Day-Kelly{ | [14].)∈ } ⊆ Proposition. (1) The class exp contains all isomorphisms, is closed under composition and stableF under pullback. (2) If g f exp with g monic, then f exp. (3) If g · f ∈ F exp with f an -biquotient∈ map,F then g exp. · ∈ F F ∈ F Proof. Statements (1) and (2) are trivial, see Prop. 3.2. A pullback of g f with · f an -biquotient map has the form g0 f 0 with f 0 and -(bi)quotient map and F exp · F g0 f 0 . Hence it suffices to show that g reflects -quotients if g f does, with· f∈an F -quotient map. But if r : W Z is an -quotientF map, we can· form the consecutiveF pullback diagram → F

f 0 g0 U / V / W (39)

q p r f g X / Y / Z 10. Exponentiable maps, exponentiable spaces 151

and have that q is an -quotient map, by hypothesis on g f. But then also F · f q = p f 0 is an -quotient map, whence p is one too, by the Lemma and · · F Exercise 2 of 1.4.

10.4. Exponentiable objects. An object X of is -exponentiable if the unique X F morphism !X : X 1 is -exponentiable. This means precisely that for every -quotient map p :→W VFalso F → 1X p : X W X V × × → × is an -quotient map. From Corollaries 9.4 and 9.5 (or from 9.1) we have: F Corollary. Let be fibre-determined. Then every -compact Hausdorff object is -exponentiable.X Even every locally -compact HausdorffF object has this property, ifF the notions of +-quotient map andF -quotient map are equivalent. 2 F F Just as Theorem 3.3 follows from Prop. 3.2, one can conclude the following Theorem from Prop. 10.3:

Theorem. (1) For any -exponentiable f : X Y with Y -exponentiable, also X is -exponentiable.F → F (2) ForF any -biquotient map f : X Y with X -exponentiable, also Y is -exponentiable.F → F (3) WithF X and Y also X Y is -exponentiable. × F 2

10.5. Exponentiability in op. It is time to shed more light on the notion of exponentiability in case T = op. A characterization in traditional topological terms is, however, not quiteX obvious,T and to a large extent we must refer the interested Reader to the literature. Reasonably manageable is the case of a subspace embedding.

Proposition (Nieﬁeld [38]). For a subspace A of a topological space X, the following are equivalent: (i) the inclusion map A, X is exponentiable; (ii) A is open in its closure→ A in X; (iii) A is locally closed in X, that is: A = O C, for an open set O and a closed set C in X. ∩

Proof (Richter [44]). The equivalence (iii) (ii) is straightforward, while (ii) (i) follows easily from the characterization of⇔ quotients in op via open subsets.⇒ T (i) (ii): Let the inclusion map A, X be exponentiable. Then the inclusion f : ⇒A, A := Y is exponentiable by→ Prop. 10.3(2). Consider the inclusion of the → 152 III. A Functional Approach to General Topology

complement g : Y A, Y and the pushout (j1, j2 : Y Y +Y A Y ) of (g, g). We therefore have the\ following→ diagram → \

g Y A / Y (40) \ i2 oo 3 ooo 33 wooo 3 g 3 Y + Y j2 33 i1 r9 OOO q 3 rrr OOO 33 rrr ' 3 1Y Y V / Y +Y A Y 33 VVV j1 \ 3 VVVV HH 3 VVV H g 3 VVV H ∇ 3 VVVV HH 3 1Y VVV HH 3 VVVV H 3 VVVVHH3 VV$+ Y where q and g are quotient maps, since q is the coequalizer of (i1 g, i2 g) and ∇ · · g is by construction a split epimorphism. Form now the pullback diagrams ∇ r p A + A / B / A (41)

s f

q g Y + Y / Y +Y A Y ∇ / Y \ where s : B, Y +Y A Y is an inclusion and the map r : A+A B is an identity. Since with f→also s is\ exponentiable, then r : A + A B is a→ quotient, hence an → homeomorphism. The inclusion k1 : A, A+A = B is open, and therefore there is → ∼ 1 an open subset O of Y +Y A Y such that O (A + A) = k1(A), hence j1− (O) A. 1 \ 1 ∩ 1 ⊇ Moreover, j2− (O) = = j1− (O) (Y A) since otherwise j2− (O) A = because ∅ ∩ 1 \ ∩ 6 ∅ A is dense in Y . Therefore A = j1− (O) is open in Y as claimed. For the characterization of exponentiable spaces we refer to [14, 26, 21]: Theorem (Day-Kelly). A topological space is exponentiable if and only if for every neighbourhood V of a point there is a smaller neighbourhood U such that every open cover of V contains a finite subcover of U. 2 Spaces satisfying the condition of the Theorem are called core-compact. These are precisely the spaces whose system of open sets forms a continuous lattice (see [47, 26]). We already know that locally compact Hausdorff spaces are of this type. But in fact, without any separation condition, already Brown [4] proved that local compactness (in the sense of Example 1 of 8.1) implies exponentiability. The converse proposition in case of a Hausdorff space goes back to Michael [37]; actually, the separation condition may be eased from Hausdorff to sober, see [24]. But in general, exponentiable spaces fail to be locally compact, although no constructive example is known (see Isbell [26]). Niefield [38] established a characterization of exponentiable maps in op in the Day-Kelly style and thereby greatly generalized the Theorem above.T For recent accounts of this characterization we refer the Reader to Niefield [40] and Richter 10. Exponentiable maps, exponentiable spaces 153

[45], with the latter paper giving a smooth extension of the Day-Kelly result in terms of ﬁbrewise core-compactness. exp 10.6. Local separatedness. We brieﬂy look at morphisms in the class ( )0. F Hence, we call f : X Y locally -separated if δf : X X Y X is - exponentiable. With Prop.→ 4.2 appliedF to exp in lieu of and→ with Prop.× 10.3(3)F we obtain: F F Proposition. (1) The class of locally -separated maps contains all monomorphisms, is closed under compositionF and stable under pullback. (2) If g f is locally -separated, so is f. (3) If g · f is locally F -separated with an -exponentiable -biquotient map f, also· g is locallyF -separated. F F F 2

Example. The terminology justifies itself in case = op. With Prop. 10.5 one easily verifies that f : X Y is locally ( -)separatedX T if and only if every point → F in X has a neighbourhood U such that f U : U Y is separated. Both separated and locally injective maps (see Exercise 1| of 4.1)→ are locally separated. An object X of is locally -separated (or locally -Hausdorff) if X 1 is X F F → locally -separated, i.e., if δX : X X X is -exponentiable. For = op, this meansF that every point in X has→ a Hausdorff× F neighbourhood. X T Exercises. 1. Apply Theorem 4.3 and Corollary 4.3 with exp in lieu of in order to establish stability properties for locally -separated spaces.F (Attention:F special care needs to be given when “translating” conditionF (vi) of Theorem 4.3: see (3) of the Proposition above.) 2. Show that in op locally Hausdorff spaces are sober T1-spaces, but not conversely. T 10.7. Maps with exponentiable domain and locally separated codomain. A pow- erful sufficient criterion for -exponentiability of morphisms (a weaker version of which was established in [39])F is obtained “for free” when we apply Prop. 5.1 to exp in lieu of : F F Theorem. Every morphism f : X Y in with X -exponentiable and Y locally -Hausdorff is -exponentiable. → X F 2 F F Just as Prop. 5.2 follows from Prop. 5.1, we conclude from the Theorem the following strengthening of the assertion of Prop. 10.3(2): Corollary. If g f is -exponentiable with g locally -separated, then f is - exponentiable. · F F F

Exercise. Prove that the full subcategory of -exponentiable and locally -separated F F objects is closed in under ﬁnite limits and under -exponentiable subobjects. X F 154 III. A Functional Approach to General Topology

10.8. Adjoints describing exponentiability. We saw in 10.2 that f : X Y is → -exponentiable in op if and only if f ∗ : op/Y op/X has a right adjoint functor.F It is a slightlyT tricky exercise on adjointT functors→ T to show that the latter property may equivalently be expressed by the adjointness of two other functors (see Nieﬁeld [38]): Proposition. In a ﬁnitely-complete category , the following assertions are equivalent for a morphism f : X Y : X → (i) f ∗ : /Y /X has a right adjoint; X → X (ii) the functor X Y ( ): /Y , (V, p) X Y V , has a right adjoint; × − X → X 7→ × (iii) the functor (X, f) ( ): /Y /Y , (V, p) (X Y V, f f ∗(p)), has a right adjoint.× − X → X 7→ × ·

Proof. Note that the functor described by (ii) is simply ΣX f ∗, with the forgetful ΣX : /X , and the functor described by (iii) may be written as f!f ∗, where X → X f! : /X /Y, (U, q) (U, f q), X → X 7→ · is left adjoint to f ∗. Property (iii) identiﬁes (X, f) as an exponentiable object of the category /Y since, in general, one calls an object X of a category with ﬁnite products expo-X nentiable if X ( ): has a right adjoint. X × − X → X 10.9. Function space topology. As important as establishing the existence of the right adjoints in question is their actual description. In case = op, the right adjoint to X ( ) for an exponentiable (=core-compact, see 10.5)X T space is described by the function-space× − functor ( )X : op op, − T → T where the space Y X has underlying set C(X,Y ) = op(X,Y ) (the set of continuous functions from X to Y ); its open sets are generatedT by the sets 1 N(U, V ) = f C(X,Y ) U <

10.10. Partial products. How does the function-space functor look like in op/Y , T rather than in op? Here we just indicate how the right adjoint of X Y ( ): op/Y op T(see Prop. 10.8(ii)) looks like for f : X Y exponentiable× in −op. SuchT right→ T adjoint functor must produce, for every object→ Z in op, an objectT T (P, p : P Y ) in op/Y and a morphism ε : X Y P Z which makes → T × → ε Z o X Y P / P (42) × p f X / Y terminal amongst all diagrams

d Z o X Y T / T (43) × t f X / Y such that there is a unique morphism h : T P with p h = t and ε (1X h) = d (see [19]). Exploiting the bijective correspondence→ · · × h : T P → t : T Y, d : X Y T Z → × → in case T = 1, one sees that P = P (f, Z) should have underlying set 1 P = (t, d) t Y, d : f − t Z continuous , { | ∈ → } with projection p : P Y , and that ε = εf,Z is the evaluation map (x, d) d(x) → 7→ when we realize the underlying set of the pullback X Y P as × 1 X Y P = (x, d) x X, d : f − (f(x)) Z continuous . × { | ∈ → } Now, for f exponentiable, there is a coarsest topology on P which makes both p and ε continuous. This topology has been described quite succinctly in terms of ultrafilter convergence in [10]. The case that f : U, Y is the embedding of an open subspace of Y (hence exponentiable, by Prop. 10.5)→ deserves special mentioning. In this case, the underlying sets of P and U Y P may be realized as (U Z) + (Y U) and U Z, × × \ × respectively, with p U Z and ε projection maps and p Y U the inclusion map. Here the coarsest topology| × on P making p and ε continuous| \ provides the pullback U Y P = U Z with the product topology, and it was first described by Pasynkov [41]× who called× P the partial product of Y over U with fibre Z. He gave various interesting examples for such spaces; for instance, the n-dimensional sphere Sn can be obtained recursively via partial products from the n-dimensional cube In and the discrete doubleton D2, as n n n 1 n S = P (I S − , I ,D2). ∼ \ → 156 III. A Functional Approach to General Topology

Exercises. 1. Identify the functors described by Prop. 10.8 in the commutative diagram /X (44) X; F xx FF xx FF xx FF xx F# /Y / /Y X GG wX GG ww GG ww GG ww ΣY G# {ww X 2. Show that Z (Y Z,Y Z Y ) is right adjoint to ΣY . 7→ × × → 3. Conclude that when we denote by ( )(X,f) the right adjoint of (X, f) ( ) in op/Y , − × − T then P (f, Z) = (Y Z,Y Z Y )(X,f). ∼ × × →

11. Remarks on the Tychonoﬀ Theorem and the Stone-Cechˇ com- pactiﬁcation

11.1. Axioms giving closures. If is op or oc, in addition to (F3)-(F5) the class satisﬁes also the followingX condition:T L F (F6) arbitrary intersections of morphisms in exist and belong to . F ∩M F ∩M Hence, is stable under intersection (see Prop. 1.4(4); we allow the indexing systemF to ∩ be M arbitrarily large). Equipped with this additional condition in our category satisfying the conditions of 2.1, one introduces the -closure X F cX (m): cX (M) X → of a subobject m : M X in X by → cX (m) := k subX k m, k -closed . ^{ ∈ | ≥ F } The following properties are easily checked:

1. m cX (m), ≤ 2. m n cX (m) cX (n), ≤ ⇒ ≤ 3. cX (m) ∼= cX (cX (m)), 4. f[cX (m)] cY (f[m]), ≤ for all m, n subX and f : X Y in . In other words, c = (cX )X is an idempotent∈ closure operator in→ the senseX of [17]. In what follows we therefore∈X assume that is given as in 2.5, that is: F (F7) a morphism f lies in if and only if f[ ] preserves -closed subobjects. F − F The condition for f means equivalently that image commutes with closure, i.e., in assertion 4 above we may write = instead of when f is closed. Of course, (F7) ∼ ≤ implies (F5) when is stable under pullback along morphisms in 0 (see Exercise 4 of 2.1). InE addition, we assume the closure operator c to beF hereditary∩ M , that is: 11. Remarks on the Tychonoﬀ Theorem, Stone-Cechˇ compactiﬁcation 157

1 (F8) cY (m) = y− [cX (y m)] for all m : M Y , y : Y X in . ∼ · → → M Exercises. 1. Verify that op and oc satisfy conditions (F6)-(F8). 2. Show that (F8)T impliesL that every morphism in is -initial (see 6.5). 3. Prove that when satisﬁes (F6)-(F8), so does M/B. F X X

11.2. Products as inverse limits of ﬁnite products. Every product of objects in is an inverse limit of its “ﬁnite subproducts”. This means, given a product X

X = X I Y i i I ∈ of objects in , one has X XI = lim XF , ∼ F Iﬁnite ⊆

where XF = Xi, with canonical bonding morphisms XG XF for F G I. Y → ⊆ ⊆ i F It is then natural∈ to expect that the closure operator deﬁned in 11.1 commutes with this limit presentation:

(F9) cX (M) = lim cX (πF [M]) for all m : M XI in ; I ∼ F Iﬁnite F → M ⊆ here πF : XI XF is the canonical morphism. Equivalently, this formula may be written as: →

1 c (m) = π− [c (π [m])] XI ∼ ^ F XF F F Iﬁnite ⊆

(see [9]). Hence, by (F9) the “topological structure” of XI is completely determined by the structure of the “ﬁnite subproducts”.

Exercises. 1. Verify that op and oc satisfy condition (F9). 2. Prove that whenT satisﬁesL (F6)-(F9), so does /B. X X

11.3. Towards proving the Tychonoﬀ Theorem. Let us now assume that all Xi (i I) are -compact. In order to show that the projection ∈ F p : XI Y Y × → is -closed for every object Y in , by (F7) we would have to establish F X p[c(m)] = c(p[m]) ( ) ∼ ∗ 158 III. A Functional Approach to General Topology

for all m : M XI Y in . For that consider the commutative diagram → × M c(M) (45) 7 ×× 77 ×× 77 ×× 77 ×× 77 ×× 77 ×× 77 ×× 77 ×× 77 ×× 77 × 7 e ×× XI Y 77 ×× ×?? 77 × π ? p 7 ×× F ?? 77 × ?? 7 ×× ?? 77 × pF 7 × X Y 7 ×× F / Y gO 77 × o7 × OOO 7 ×× ooo OOO 77 × ooo OOO 7 ×× oo OO 77 × ooo OOO 7 Ó× o eF c(πF [M]) / c(p[M])

Here e, eF are determined by the projections p, pF , respectively, and πF = πF 1Y × represents XI Y as an inverse limit of XF Y (F I finite). Formula ( ) says × × ⊆ ∗ precisely that e should be in . By Theorem 3.3(3), pF is -closed for every finite E F F I, hence eF . Now, by (F9), ⊆ ∈ E c(M) = lim c(πF [M]) ∼ F in , which implies X (c(M), e) = lim (c(πF [M]), eF ) ∼ F in /c(p[M]). X 11.4. The Tychonoff Theorem. We call a surjectivity class if for some class of objects in one has f : X Y in exactlyE when all maps (P, f): (P,XP) X → E X X → (P,Y )(P ) are surjective. Note that if in is a surjectivity class, so is B Xin /B. Furthermore,∈ P when is fibre-determinedE X (see 9.3), then is a surjectivityE class.X In op, is a surjectivityX class (take = 1 ), but in ocE it is not (since every surjectivityT E class is pullback stable). P { } L

Theorem (Clementino-Tholen [11]). In addition to (F1)-(F5), let satisfy conditions (F6)-(F9) of 11.1/2. Then each of the following two assumptionsX makes -Comp closed under direct products in : F X (L) if (X, πν : X Xν )ν is an inverse limit in and (eν : Xν Y )ν a compatible family→ of morphisms in , then theX induced morphism→ e : X Y lies in ; E (T) is→ a surjectivityE class, and the Axiom of Choice holds true. E 11. Remarks on the Tychonoﬀ Theorem, Stone-Cechˇ compactiﬁcation 159

Proof (sketch). By 11.3, assumption (L) makes -Comp closed under products in . Regarding assumption (T), it is suﬃcient toF deduce that for the particular X inverse system of diagram (43), eF for all ﬁnite F I implies e in . This is ∈ E ⊆ E shown (using an ordinal induction) in [11] and [9].

Examples. (1) Via (T) one obtains the classical Tychonoff Theorem in op. Note that the validity of the Tychonoff Theorem in op is logicallyT equivalent to the Axiom of Choice (see [32]). T (2) While (L) fails in op, the condition is satisfied in oc (see [49], 2.3), and the Theorem givesT a choice-free proof for the TychonoffL Theorem in oc. (See also Chapter II.) L

Exercises.

1. Assuming (F1)-(F6) show that for every monic family (pi : X Xi)i I (so that for → ∈ x, y : P X, pi x = pi y for all i I always implies x = y) Xi -Haus implies X -Haus.→ · · ∈ ∈ F 2. Show∈ F that under the assumptions of the Theorem, -CompHaus is closed under all small limits in , if has them. F X X

11.5. Products of proper maps. Since all assumptions of Theorem 11.4 are in- variant under “slicing”, we may apply it to the slices /B rather than to and obtain: X X Corollary. If satisﬁes (F1)-(F9) and condition (L) or (T) of Theorem 11.4, then the direct productX of -proper ( -perfect) morphisms in is again -proper ( - perfect, respectively).F F X F F

Proof. Given -proper ( -perfect) maps fi : Xi Yi, i I, the morphism F F → ∈ fi : Xi Yi = Y may be regarded as an object of /Y and is then a Y Y → Y X i I i I i I ∈ ∈ ∈ product of the objects (Pi, f 0), where f 0 : Pi Y is the pullback of fi along the i i → projection Y Yi, i I. → ∈ In case = op the Corollary (which formally generalizes Theorem 11.4) is known as Frolik’sX T Theorem. 11.6. Existence of the Stone-Cechˇ compactiﬁcation. A full subcategory of is said to be -cowellpowered if every object X admits only a smallA setX of non-isomorphicF -dense morphisms with domain∈X Aand codomain in ; that is: F A if there is a set-indexed family di : X Ai (i I) of -dense morphisms in → ∈ F A such that every -dense morphism d : X A in factors as d = j di for some isomorphism j andF i I. → A · An easy application∈ of Freyd’s General Adjoint Functor Theorem gives (see [35]): 160 III. A Functional Approach to General Topology

Theorem. Let be small-complete, and let -Haus be -cowellpowered. Then -CompHaus isX reflective in -Haus with F-dense reflexionsF if and only if - CompHausF is closed under productsF in . F F X Proof (sketch). The necessity of the condition is clear (see also Exercise 1 of 11.4). For its sufficiency, let X -Haus and consider a representative system di : X ∈ F → Ai (i I) of -dense morphisms as in the definition of -cowellpoweredness, with ∈ F F the additional condition that Ai -CompHaus. By (F6), the induced morphism ∈ F f : X Ai factors through the -closed subobject → Y F i I ∈

βX := c(f[X]) Ai −→ Y i I ∈ by a unique morphism βX : X βX, which is -dense by (F8). Note that βX is → F -compact Hausdorﬀ, by Theorem 3.3(3), since Ai is. F Y i I An arbitrary morphism g : X A with A ∈-CompHaus factors through the → ∈ F -dense morphism g0 : X c(g[X]), which must be isomorphic to some di and F → must therefore factor through βX . The resulting factorization is unique since βX is -dense and A is -Hausdorﬀ. F F

βX X T / βX (46) HHTTTT HH TTT f HH TTT HH TTTT HH TTT H di T) g HH A HH i I i HH Q ∈ HH H pi HH HH H$ A o c(g[X]) o ∼ Ai Of course, reflectivity of -CompHaus in -Haus gives in particular reflectivity of -CompHaus in -TychF and therefore theF existence of the Stone-Cechˇ compacti- ficationF , as neededF in Section 6. Exercise. Show that the cardinal number of the underlying set of a Hausdorff space X cardA with dense subset A cannot exceed 22 . Conclude that the full subcategory -Haus F in op is -cowellpowered, so that Theorem 11.6 in conjunction with the Tychonoff T F Theorem gives the existence of the classical Stone-Cechˇ compactification of Tychonoff spaces.

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