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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 im- portant 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 gen- erally, 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 oper- ator [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 subob- jects. 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 the- orem 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 cat- egories, 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) hy- potheses. Table 2 replicates part of Table 1 at the object level. Many of these notions get discussed throughout the chapter in standard exam- ples, 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 , andM 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 · (written2 M as e 2m E), that is: given any2 morphismsE u; v with m u =2 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 (orthog- onal) 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.
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