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D-18 Theˇcech–Stone Compactifications of N and R

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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 . Reprinted from Encyclopedia of General Topology K. P. Hart, J. Nagata and J. E. Vaughan, editors © 2004 Elsevier Science Ltd. All rights reserved 214 Section D: Fairly general properties 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.
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