D-18 Theˇcech–Stone Compactifications of N and R

d-18 The Cech–Stoneˇ compactiﬁcations of N and R 213 d-18 The Cech–Stoneˇ Compactiﬁcations of N and R

c It is safe to say that among the Cech–Stoneˇ compactifica- The proof that βN has cardinality 2 employs an in- tions of individual spaces, that of the space N of natural dependent family, which we define on the countable set numbers is the most widely studied. A good candidate for {hn, si: n ∈ N, s ⊆ P(n)}. For every subset x of N put second place is βR. This article highlights some of the most Ix = {hn, si: x ∩ n ∈ s}. Now if F and G are finite dis- striking properties of both compactifications. T S joint subsets of P(N) then x∈F Ix \ y∈G Iy is non-empty – this is what independent means. Sending u ∈ βN to the P( ) ∗ point pu ∈ N 2, defined by pu(x) = 1 iff Ix ∈ u, gives us a 1. Description of βN and N c continuous map from βN onto 2. Thus the existence of an independent family of size c easily implies the well-known The space β (aka βω) appeared anonymously in [10] as an N fact that the Cantor cube c2 is separable; conversely, if D is example of a compact Hausdorff space without non-trivial c dense in 2 then setting Iα = {d ∈ D: dα = 0} defines an converging sequences. The construction went as follows: for ∗ ∈ independent family on D. Any closed subset F of N such every x (0, 1) let 0.a1(x)a2(x)... an(x)... be its dyadic c expansion (favouring the one that ends in zeros). This gives that the restriction to F of the map onto 2 is irreducible is (homeomorphic with) the absolute of c2. If a point u of ∗ us a countable set A = {an(x): n ∈ N} of points in the Ty- N chonoff cube [0, 1](0,1). The closure of A is the required belongs to such an F then every compactification of the space ∪ {u} contains a copy of β . space. To see that this closure is indeed βN one checks N N Next we prove that ∗ is very non-separable by working that the map n 7→ an induces a homeomorphism of β N N < onto cl A. Indeed, it suffices to observe that whenever X is on the tree 2 N of finite sequences of zeros and ones. For ∈ N = { : ∈ } { : ∈ N } a coinfinite subset of N one has an(x) = 1 iff n ∈ X, where every x 2 let Ax x n n N . Then Ax x 2 is an P −n { ∗: ∈ N } x = ∈ 2 . almost disjoint family of cardinality c and so Ax x 2 n X ∗ The present-day description of β is as the Stone space of is a disjoint family of clopen sets in N . N ∗ the Boolean algebra P(N). Thus the underlying set of βN is We can use this family also to show that N is not ex- S the set of all ultrafilters on N with the family {X: X ⊆ N} as tremally disconnected. Let Q denote the points in N2 that a base for the open sets, where XS denotes the set of all ultra- are constant on a tail (these correspond to the endpoints of filters of which X is an element. The space βN is a separable the Cantor set in [0, 1]) and let P = N2 \ Q. Then OQ = S ∗ S ∗ ∗ and extremally disconnected compact Hausdorff space and x∈Q Ax and OP = x∈P Ax are disjoint open sets in N , c ∗ its cardinality is the maximum possible, i.e., 2 . yet cl OQ ∩ cl OP 6= ∅, for if A is such that Ax ⊆ A for Most of the research on βN concentrates on its remain- all x ∈ P then the Baire Category Theorem may be applied \ ∗ ∗ ∗ der βN N, which, as usual, is denoted N . By extension to find x ∈ Q with Ax ⊆ A. This example shows that is ∗ S N one writes X = X \ N for subsets X of N. The family not basically disconnected: O is an open F -set whose ∗ Q σ B = {X : X ⊆ N} is precisely the family of clopen sets closure is not open. ∗ ∗ ⊆ ∗ \ B of N . Because X Y iff X Y is finite the algebra is The algebra P(N)/fin has two countable (in)completeness isomorphic to the quotient algebra P(N)/fin of P(N) by the h i ∗ properties. The first states that when bn n is a decreasing ideal of finite sets – hence N is the Stone space of P(N)/fin. ∗ sequence of non-zero elements there is an x with bn > x > Much topological information about comes from knowl- ∗ N 0 for all n; in topological terms: nonempty Gδ-sets on N edge of the combinatorial properties of this algebra. In prac- ∗ have nonempty interiors and N has no isolated points. The P tice one works in (N) with all relations taken modulo fi- second property says that when hb i is as above and, in nite. We use, e.g., X ⊆∗ Y to denote that X \ Y is finite, n n addition, ha i is an increasing sequence with a < b for X ⊂∗ Y to denote that X ⊆∗ Y but not Y ⊆∗ X, and so on. n n n n all n there is an x with a < x < b for all n; in topological In this context the word ‘almost’ is mostly used in place of n n terms: disjoint open F -sets in ∗ have disjoint closures, ‘modulo finite’, thus ‘A and B are almost disjoint’ means σ N i.e., ∗ is an F-space. A ∩ B =∗ ∅. N We now turn to sets A and B where no interpolating x can be found, that is, we look for A and B such that W A0 < V B0 ∗ 0 <ω 0 <ω 2. Basic properties of N whenever A ∈ [A] and B ∈ [B] but for which there is no x with a 6 x 6 b for all a ∈ A and b ∈ B. The minimum ∗ Many results about N are found by constructing special cardinalities of sets like these are called cardinal character- families of subsets of N, although the actual work is often istics of the continuum and these cardinal numbers play an ∗ done on a suitable countable set different from N. important role in the study of βN and N .

We have already seen such a situation with a countable A By the Axiom of Choice every almost disjoint family can ∗ ∗ and a B of cardinality c: the families {Ax }x∈Q and {N \ be extended to a Maximal Almost Disjoint family (a MAD ∗ ∗ S ∗ Ax }x∈P . However, this case, of a countably infinite A, is best family). If A is a MAD family then N \ {A : A ∈ A} is visualized on the countable set N × N. For n ∈ N put an = nowhere dense and every nowhere dense set is contained in ∗ ∗ (n×N) and for f ∈ NN put b f = {hm, ni: n > f (m)} ; then such a ‘canonical’ nowhere dense set. The minimum num- ∗ A = {an: n ∈ N} and B = {b f : f ∈ NN} are as required: if ber of nowhere dense sets whose union is dense in N is de- an < x for all n then one readily finds an f such that b f < x noted h and is called the weak Novák number. It is equal to (make sure {n} × [ f (n), ∞) ⊂ x). A pair like (A, B) above the minimum number of MAD families without a common is called an (unfillable) gap; gap, because, as in a Dedekind MAD refinement – in Boolean terms, it is the minimum κ gap, one has a < b whenever a ∈ A and b ∈ B, and unfillable for which P(N)/fin is not (κ, ∞)-distributive. because there is no x such that a 6 x 6 b for all a and b. Interestingly [1], one can find a sequence hAα: α < hi of Because unfillable gaps are the most interesting one drops MAD families, without common refinement and such that the adjective and speaks of gaps. (1) Aβ refines Aα whenever α < β; (2) if A ∈ Aα then {B ∈ N ∗ One does not need the full set N to create a gap; it Aα+1: B ⊆ A} has cardinality c; and (3) the family T = suffices to have a subset U such that for all g ∈ N there S ∗ N α Aα is dense in P(N)/fin – topologically: {A : A ∈ T } is f ∈ U such that {n: f (n) > g(n)} is infinite. The mini- is a π-base. This all implies that, with hindsight, there is an mum cardinality of such a set is denoted b. increasing sequence of closed nowhere dense sets of length h The properties of b and its bigger brother d are best ex- whose union is dense and that h is a regular cardinal; also ∗ plained using the relation < on NN where, in keeping note that T is a tree under the ordering ⊃∗. ∗ with the notation established above, f < g means that One can use such a tree, of minimal height, in induc- {n: f (n) g(n)} is finite. The definition of b given above ∗ > tive constructions, e.g., as in [2] to show that N is very identifies it as the minimum cardinality of an unbounded – ∗ non-extremally disconnected: every point is a c-point, which with respect to < – subset of NN; unbounded is not the means that one can find a family of c many disjoint open sets same as dominating (cofinal), the minimum cardinality of a each of which has the point in its closure. An important open dominating subset is denoted d and it is called the dominat- problem, at the time of writing, is whether this can be proved ing number. for every nowhere dense set, i.e., if for every nowhere dense × If one identifies N and N N, as above, then one recog- subset of ∗ one can find a family of c disjoint open sets = S ∗ N nizes d as the character of the closed set F cl n An and each of which has this set in its boundary – in short, whether b as the minimum number of clopen sets needed to create every nowhere dense set is a c-set. ∗ \ an open subset of N F whose closure meets F. In either The latter problem is equivalent to a purely combina- characterization of b the defining family can be taken to be torial one on MAD families: for every MAD family A well-ordered. the family I+(A) should have an almost disjoint refine- In case where A is finite one can simplify matters by let- ment. Here I(A) is the ideal generated by A and I+(A) = ting A = {0}. If B is an ultrafilter then there is no x with P( ) \ I(A); an almost disjoint refinement of a fam- 0 < x < b for all b ∈ B, hence there are filters whose only N ily B is an indexed almost disjoint family {A : B ∈ B} with lower bound is 0; the minimum cardinality of a base for such B A ⊆∗ B for all B. Another characterization asks for enough a filter is denoted p. Alternatively one can ask for chains B (on even one) MAD families of true cardinality c, i.e., if without positive lower bound: the minimum length of such a X ∈ I+(A) then X ∩ A 6=∗ ∅ for c many members of A. chain is denoted t – it is called the tower number. A defin- If the minimum size of a MAD family, denoted a, is equal ing family for the cardinal t can be used to create a sepa- to c then this is clearly the case, however there are various rable, normal, and sequentially compact spaces that is not models with a < c. compact. When we let A become uncountable we encounter two important types of objects of cardinality ℵ1: Hausdorff gaps and Lusin families. A Hausdorff gap [5] is a pair of se- 3. Homogeneity quences A = haα: α < ω1i and B = hbα: α < ω1i of elements such that aα < aβ < bβ < bα whenever α < β but for Given that the algebra P(N)/fin is homogeneous it is some- ∗ which there is no x with aα < x < bα for all α.A Lusin what of a surprise to learn that the space N is not a homo- family [7] is an almost disjoint family A of cardinality ℵ1 geneous space, i.e., there are two points x and y for which ∗ such that no two uncountable and disjoint subfamilies of A there is no autohomeomorphism h of N with h(x) = y. can be separated, i.e., if B and C are disjoint and uncount- The first example of this phenomenon is from [9]: the ∗ ∗ ∗ able then there is no set X such that B ⊆ X and X ∩C = ∅ Continuum Hypothesis (CH) implies that N has P-points; for all B ∈ B and C ∈ C. One can parametrize the F-space a P-point is one for which the family of neighbourhoods is ∗ property: an Fκ -space is one in which disjoint open sets that closed under countable intersections. Clearly N has non-P- are the union of fewer than κ many closed sets have dis- points (as does every infinite compact space), so CH implies ∗ ∗ joint closures. The two families above show that N is not N is not homogeneous. To construct P-points one does not = an Fℵ2 -space. need the full force of CH, the equality d c suffices. d-18 The Cech–Stoneˇ compactifications of N and R 215

∗ ∗ ∗ ∗ In [4] one finds a proof of the non-homogeneity of N that P-set in N , viz. (N × {ω}) , that is homeomorphic to N ∗ avoids CH; it uses the Rudin–Frolík order on N , which itself. ∗ is defined by: u @ v iff there is an embedding f : βN → N Parovicenko’sˇ ‘other theorem’ states that every compact ∗ such that f (u) = v. If u @ v then there is no autohomeomor- space of weight ℵ1 (or less) is a continuous image of N , ∗ ∗ phism of N that maps u to v. This proof works to show that whence under CH the space N is a universal compactum of no compact F-space is homogeneous: if X is such a space weight c, in the mapping onto sense. ∗ then one can embed βN into it, no autohomeomorphism of X Virtually everything known about N under CH follows can map (the copy of) u to (the copy of) v. from Parovicenko’sˇ theorems. To give the flavour we show That this proof avoiding CH was really necessary be- that a compact zero-dimensional space can be embedded ∗ ∗ came clear when the consistency of “N has no P-points” into N if (and clearly only if) it is an F-space of weight c was proved. In [6] the P-point proof was salvaged partially: (or less). Indeed, let X be such a space and take a P-set A ∗ ∗ ∗ N has weak P-points, i.e., points that are not accumulation in N that is homeomorphic to N and a continuous onto points of any countable subset. map f : A → X. This induces an upper-semi-continuous de- ∗ ∗ Though not all points of N are P-points one may still composition of N whose quotient space, the adjunction try to cover ∗ by nowhere dense P-sets. Under CH this ∗ N space N ∪ f X, can be shown to have all the properties that ∗ ∗ cannot be done but it is consistent that it can be done. The characterize N , hence it is N and we have embedded X ∗ ∗ principle NCF implies that N is the union of a chain, of into N , even as a P-set. length d, of nowhere dense P-sets. The principle NCF (Near In the absence of CH very few of the consequences of Coherence of Filters) says that any two ultrafilters on N are Parovicenko’sˇ theorems remain true. The characterization ∗ nearly coherent, i.e., if u, v ∈ N then there is a finite-to- theorem is in fact equivalent to CH and for many concrete one f : → such that β f (u) = β f (v). ∗ ∗ ∗ ∗ N N spaces, like N ×N , R , (N×(ω+1)) and the Stone space This in turn implies that the Rudin–Keisler order is of the measure algebra it is not a theorem of ZFC that they downward directed; we say u ≺ v if there is some f : → ∗ N N are continuous images of N . It is also consistent with ZFC = ∗ such that u f (v). This is a preorder on βN and a partial that all autohomeomorphisms of are trivial, i.e., induced ≺ ≺ N order on the types of βN: if u v and v u then there is by bijections between cofinite sets. This is a far cry from the a permutation of N that send u to v. The Rudin–Frolík and 2c autohomeomorphisms that we got from CH. It is also con- Rudin–Keisler orders are related: u v implies u ≺ v, so ∗ @ @ sistent that N cannot be homeomorphic to a nowhere dense is a partial order on the types as well. P-subset of itself; this leaves open a major question: is there Both orders have been studied extensively; we mention ∗ a non-trivial copy of N in itself, i.e., one not of the form some of the more salient results. There are -minimal ∗ @ cl D \ D for some countable discrete subset of N . points in ∗: weak P-points are such. Points that are ≺- N A good place to start exploring β is van Mill’s survey minimal in ∗ are called selective ultrafilters; they exist if N N [KV, Chapter 11]. CH holds but they do not exist in the random real model. There are ≺-incomparable points (even a family of 2c many ≺-incomparable points) but it is not known whether for every point there is another point ≺-incomparable to it. The 5. Cardinal numbers order @ is tree-like: types below a fixed type are linearly ordered. The cardinal numbers mentioned above are all uncountable and not larger than c, so the Continuum Hypothesis (CH) implies that all are equal to ℵ1. More generally, Martin’s Ax- 4. The continuum hypothesis and βN iom implies all are equal to c. One can prove certain inequalities between the characteristics, e.g., ℵ1 6 p 6 t 6 h 6 b 6 ∗ = The behaviour of βN and, especially, N under the assump- d. Intriguingly it is as yet unknown whether p t is prov- tion of the Continuum Hypothesis (CH) is very well un- able, what is known is that p = ℵ1 implies t = ℵ1. derstood. The principal reason is that the Boolean algebra One proves b 6 a but neither a 6 d nor d 6 a is provable P(N)/fin is characterized by the properties of being atom- in ZFC. less, countably saturated and of cardinality c = ℵ1. Topo- Two more characteristics have received a fair amount of ∗ logically, N is, up to homeomorphism, the unique com- attention, the splitting number s is the minimum cardinal- pact zero-dimensional without isolated points, which is an ity of a splitting family, that is, a family S such that for every F-space in which nonempty Gδ-sets have non-empty inte- infinite X there is S ∈ S with X ∩ S and X \ S infinite; and rior and which is of weight c. This is known as Parovicenko’sˇ its dual, the reaping number r, which is the minimum car- ∗ characterization of N and it implies that whenever X is dinality of a family that cannot be reaped (or split), that is, a compact zero-dimensional and of weight c (or less) the family R such that for every infinite X there is R ∈ R such ∗ ∗ remainder (N × X) is homeomorphic to N . This pro- that one of X ∩ R and X \ R is not infinite. ∗ c ∗ κ vides us with many incarnations of N , e.g., as (N × 2) , Topologically s is the minimum κ for which 2 is not se- which immediately provides us with 2c many autohomeo- quentially compact and r is equal to the minimum π-cha- ∗ ∗ ∗ morphisms of N , or as (N × (ω + 1)) , which gives us a racter of points in N . 216 Section D: Fairly general properties

∗ Further inequalities between these characteristics are, e.g., cut point (i.e., cut point of some subcontinuum) of H . Al- h 6 s 6 d and b 6 r. though the continuum Iu is an irreducible continuum, i.e., These ‘small’ cardinals are the subject of ongoing re- no smaller continuum contains its end points au and bu , it search; good introductions are [KV, Chapter 3] and [vMR, is definitely not an ordered continuum. It is an F-space so Chapter 11]. the closure of an increasing sequence of points in the ultraproduct is homeomorphic to βN (in an ordered continuum it would have to be ω+1). The ‘supremum’ of such a sequence 6. Properties of β R of points in Iu is an irreducibility layer of Iu ; this layer is ∗ non-trivial (it contains N ) and indecomposable. Adding all Instead of βR one usually considers βH, where H is the these facts together we can deduce that a maximal chain of positive half line [0, ∞). This is because x 7→ −x induces ∗ indecomposable subcontinua of H has a one-point inter- an autohomeomorphism of β that shows that β[0, ∞) and ∗ R section; such a point is not a weak cut point of H . Thus β(−∞, 0] are the same thing. ∗ H is shown to be not homogeneous by purely continuum- In a sense βH looks like βN in that it is a thin locally theoretic means. There is a natural quasi-order on an irre- compact space with a large compact lump at the end; this re- ducible continuum: in the present case x 6 y means “every mainder ∗ has some properties in common with ∗: both H N continuum that contains au and y also contains x”. An irre- are F-spaces in which nonempty Gδ-sets have nonempty in- ducibility layer is an equivalence class for the equivalence terior, both have tree π-bases and neither is an Fℵ2 -space. ∗ ∗ relation “x 6 y and y 6 x”. Under CH the spaces H and N have homeomorphic dense The weak cut points constructed above are near points sets of P-points. That is where the superficial similarities ∗ and, in fact, every near point is a weak cut point of . Un- end because β and ∗ are connected and β and ∗ most H H H N N der CH one can construct different kinds of weak cut points: certainly are not. far but not remote and even remote. Under CH it is also pos- A deeper similarity is a version of Parovicenko’sˇ univer- sible to map a remote weak cut point to a near point by an sality theorem: every continuum of weight ℵ (or less) is a 1 autohomeomorphism of ∗. On the other hand it is consis- continuous image of ∗, so that, under CH, the space ∗ is a H H H tent that all weak cut points are near points and hence that universal continuum of weight c. A version of Parovicenko’sˇ ∗ ∗ the far points are topologically invariant in H . A similar characterization theorem for H is yet to be found. ∗ consistency result for remote points is still wanting. The structure of H as-a-continuum is very interesting. ∗ ∗ The composant of a point x of H meets N in the ul- Most references for what follows can be found in [HvM, S ∗ trafilter {U: x ∈ cl n∈U [n, n + 1]} and two points from N Chapter 9]. ∗ are in the same composant of H iff they are nearly coher- The following construction is crucial for our understand- ∗ ∗ ent. Therefore, the structure of the set of composants of ing of the structure of the continuum H : take a discrete se- H is determined by the family of finite-to-one maps of to it- quence h[an, bn]in of non-trivial closed intervals and an ul- N self. This implies that the number of composants may be 2c trafilter u on N. The intersection (e.g., if CH holds), 1 (this is equivalent to the NCF principle) \ [ or 2 (in other models of set theory); whether other numbers I = cl [a b ] n, n are possible is unknown. U∈u n∈U The number of (homeomorphism types of) subcontinua ∗ is a continuum, often called a standard subcontinuum of H is as yet a function of Set Theory: in ZFC one can ∗ establish the lower bound of 14. Under CH there are ℵ of H . What is striking about this construction is not its sim- 1 plicity but that these (deceptively) simple continua govern types even though in one respect CH act as an equalizer: ∗ the continuum-theoretic properties of H . Every proper sub- CH is equivalent to the statement that all standard subcon- continuum (in particular every point) is contained in a stan- tinua are mutually homeomorphic. Most of the ZFC continua dard subcontinuum – it is in fact the intersection of some are found as intervals in an Iu with points and non-trivial family of standard subcontinua. From this one shows that layers at their ends and with varying cofinalities. ∗ H is hereditarily unicoherent – every finite intersection of standard subcontinua is a standard subcontinuum or a point – and indecomposable – standard subcontinua are nowhere References dense. From the other direction every subcontinuum is also the [1] B. Balcar, J. Pelant and P. Simon, The space of ultrafil- union of a suitable family of standard subcontinua. Thus, no ters on N covered by nowhere dense sets, Fund. Math. ∗ subcontinuum of H is hereditarily indecomposable, as stan- 110 (1980), 11–24. dard subcontinua have cut points. Indeed, Iu contains the [2] B. Balcar and P. Vojtáš, Almost disjoint refinement of Q ultraproduct n[an, bn]/u, as a dense set: the equivalence families of subsets of N, Proc. Amer. Math. Soc. 79 class of a sequence hxnin corresponds to its own u-limit, de- (1980), 465–470. noted xu . The subspace topology of this ultraproduct coin- [3] W.W. Comfort and S. Negrepontis, The Theory of Ul- cides with its order topology and every point of the ultra- trafilters, Grundlehren Math. Wiss. Einzeldarstellun- product (except au and bu ) is a cut point of Iu and so a weak gen, Band 211, Springer, Berlin (1974). d-18 The Cech–Stoneˇ compactifications of N and R 217

[4] Z. Frolík, Sums of ultraﬁlters, Bull. Amer. Math. Soc. inal: Ob odnom universal’nom bikompakte vesa ℵ, 73 (1967), 87–91. Dokl. Akad. Nauk SSSR 150 (1963), 36–39. [5] F. Hausdorff, Summen von ℵ1 Mengen, Fund. Math. 26 [9] W. Rudin, Homogeneity problems in the theory of Cechˇ (1936), 241–255. compactiﬁcations, Duke Math. J. 23 (1956), 409–419, ∗ [6] K. Kunen, Weak P-points in N , Topology, Vol. II, 633. Proc. Fourth Colloq., Budapest, 1978, Á. Császár, ed., [10] A. Tychonoff, Über einen Funktionenraum, Math. North-Holland, Amsterdam (1980), 741–749. Ann. 111 (1935), 762–766. [7] N.N. Luzin, O chastyakh natural’nogo ryada, Izv. Akad. Nauk SSSR. Ser. Mat. 11 (1947), 403–410. [8] I.I. Parovicenko,ˇ A universal bicompact of weight ℵ, Alan Dow and Klaas Pieter Hart Soviet Math. Dokl. 4 (1963), 592–595, Russian orig- Charlotte, NC, USA and Delft, The Netherlands