CONVEXITY STRUCTURES IN ZERO-DIMENSIONAL COMPACT SPACES

WIESŁAW KUBIŚ AND ANDRZEJ KUCHARSKI

Abstract. We investigate some properties of compact zero-dimensional spaces with additional convexity structures. As a main result, we prove that every retract of a Cantor cube has a binary subbase closed under the complements.

Introduction

We show that every retract of a Cantor cube, called a zero-dimensional Dugundji space, has a binary subbase closed under the complements. This generalizes the result of Heindorf [3] that zero-dimensional Dugundji spaces are supercompact and admit a binary subbase consisting of clopen sets. Introducing a suitable convexity (see the definitions below) we construct a binary subbase consisting of convex sets, which is closed under the complements. We also state and use some properties of inverse systems of topological convexity spaces. The proof of our main result is simpler than Heindorf’s one and does not require algebraic or lattice structures.

1. Preliminaries

By a geometrical space we mean a set X together with a collection G ⊂ P(X) such that:

(1) ∅, X ∈ G,

(2) ⋂ A ∈ G for nonempty A ⊂ G,

(3) If A ⊂ X and for every a, b ∈ A there exists a G ∈ G with a, b ∈ G ⊂ A then A ∈ G.

A collection G satisfying these conditions is called an interval convexity, see Calder [1]. Elements of G will be called convex sets. A halfspace is a convex set with the convex complement. The segment joining a, b ∈ X is defined by [a, b] = ⋂{G ∈ G : a, b ∈ G}. Note that G ∈ G iff for every a, b ∈ G it holds [a, b] ⊂ G. Consequently, an interval convexity is determined by its segments. If (X, G) is a geometrical space and M ⊂ X then G_{M} = {A ⊂ M : ∀ a, b ∈ A, [a, b] ∩ M ⊂ A} is easily seen to be an interval convexity in M, called the subspace convexity. Clearly, G∩M ∈ G_{M} whenever G ∈ G and the segment joining points a, b in M equals [a, b] ∩ M. If (X, G) and (Y, H) are two geometrical spaces then the product X ×Y is a geometrical space with the interval convexity consisting of all sets G×H where G ∈ G, H ∈ H (see [7, p. 14]). Note that [⟨a_{1}, a_{2}⟩, ⟨b_{1}, b_{2}⟩] = [a_{1}, b_{1}] × [a_{2}, b_{2}] for each ⟨a_{1}, a_{2}⟩, ⟨b_{1}, b_{2}⟩ ∈ X×Y. Let X and Y be two geometrical spaces. A map f: X → Y is called convexity preserving, cp-map for short, provided f^{−1}(G) is convex in X for each convex set G ⊂ Y. This is equivalent to the condition f([a, b]) ⊂ [f(a), f(b)] for each a, b ∈ X, see [7, p. 15]. For the study of convexity theory we refer to van de Vel’s monograph [7]. A Boolean median space is a triple (X, T , G) where (X, G) is a geometrical space, (X, T ) is a compact topological space and the following conditions are satisfied:

1991 Mathematics Subject Classification. Primary: 54B35, 54A05; Secondary: 52A01, 54D30.

Key words and phrases. inverse system, interval convexity, Boolean median space, continuous cp-map, binary subbase.

(M1) For each two distinct points a, b ∈ X there exists a clopen halfspace H ⊂ X such

that a ∈ H and b ∉ H. (M2) For each a, b, c ∈ X the set [a, b] ∩ [a, c] ∩ [b, c] is nonempty.

The condition (M1) implies that the intersection [a, b] ∩ [a, c] ∩ [b, c] consists of a single point, called the median of a, b, c and denoted by m(a, b, c). Let X be a Boolean median space. A subset M ⊂ X is median-stable [7, p. 121] provided m(a, b, c) ∈ M whenever a, b, c ∈ M. Every closed median-stable subset of X with the subspace topology and the subspace convexity is also a Boolean median space. Observe that the product of two Boolean median spaces is also a Boolean median space. Every Boolean median space is a topological median space in the sense of van de Vel [7, p. 269] and it is a zero-dimensional compact Hausdorff topological space. The following fact will be essential for us. Namely, the collection of all clopen halfspaces in a Boolean median space forms a binary closed subbase for its topology (cf. van Mill [5, Thm. 1.3.3] or van de Vel [7, Thm. II.1.7]). Recall that a nonempty family of sets is binary if each its subcollection with empty intersection contains two disjoint sets. Let Σ be a directed partially ordered set and let S = {X_{σ}, p_{σ}^{τ}, Σ} be an inverse system of sets such that each X_{σ} is a geometrical space and p_{σ}^{τ}’s are cp-maps, i.e. p_{σ}^{τ}: X_{τ} → X_{σ} and p_{σ}^{τ}p_{τ}^{μ} = p_{σ}^{μ} whenever σ ⩽ τ ⩽ μ. The system S will be called an inverse system of geometrical spaces. Let \underleftarrow{lim} S be the inverse limit of S in the category of sets, i.e. the set consisting of all points x ∈ ∏_{σ∈Σ} X_{σ} such that p_{σ}^{τ}(x_{τ}) = x_{σ} for all σ ⩽ τ. Denote by p_{σ} the projection of \underleftarrow{lim} S into X_{σ}. The proof of the following proposition is straightforward and therefore will be omitted.

Proposition 1.1. Let S = {X_{σ}, p_{σ}^{τ}, Σ} be an inverse system of geometrical spaces. There exists an interval convexity G in \underleftarrow{lim} S with the following properties:

(a) The geometrical space (\underleftarrow{lim} S, G) is the inverse limit in the category of geometrical spaces. In other words, if Y is a geometrical space and {f_{σ}: Y → X_{σ}}_{σ∈Σ} is a collection of cp-maps such that p_{σ}^{τ}f_{τ} = f_{σ} for σ ⩽ τ then there exists a unique cp-map h: Y → X with the property p_{σ}h = f_{σ} for all σ ∈ Σ.

(b) G is the least convexity in \underleftarrow{lim} S such that all the projections p_{σ} are cp-maps.

(c) For every a, b ∈ \underleftarrow{lim} S it holds [a, b]_{G} = X ∩ ∏_{σ∈Σ}[a_{σ}, b_{σ}].

By an inverse system of Boolean median spaces we mean an inverse system S = {X_{σ}, p_{σ}^{τ}, Σ} where X_{σ}’s are Boolean median spaces and each p_{σ}^{τ} is cp and continuous.

Proposition 1.2. The inverse limit of a system of Boolean median spaces is a Boolean median space.

Proof. Let S = \{X_{σ}, p_{σ}^{τ}, Σ\} be an inverse system of Boolean median spaces. Let p_{σ}: \underleftarrow{lim} S → X_{σ} be the projection. The collection

{p_{σ}^{−1}(H) : σ ∈ Σ, H is a clopen halfspace in X_{σ}}

is point-separating and consists of clopen halfspaces in \underleftarrow{lim} S. This shows (M1).

Fix a, b, c ∈ \underleftarrow{lim} S. For each σ ∈ Σ let x_{σ} = m(a_{σ}, b_{σ}, c_{σ}). If τ ⩽ σ then p_{τ}^{σ}(x_{σ}) = m(a_{τ} , b_{τ} , c_{τ}) since p_{τ}^{σ} is cp. Hence p_{τ}^{σ}(x_{σ}) = x_{τ}. It follows that x = \{x_{σ}\}_{σ∈Σ} ∈ \underleftarrow{lim} S and x ∈ [a, b] ∩ [a, c] ∩ [b, c]. This shows (M2). □

2. Some properties of Boolean median spaces

Fundamental examples of Boolean median spaces are Cantor cubes. Specifically, for a Cantor cube {0, 1}^{κ} we call its subset M convex iff

{x ∈ {0, 1}^{κ} : a^{−1}(1) ∩ b^{−1}(1) ⊂ x^{−1}(1) ⊂ a^{−1}(1) ∪ b^{−1}(1)} ⊂ M,

for every a, b ∈ M (cf. [7, p. 60]). The set H_{α} = {x ∈ {0, 1}^{κ} : x(α) = 1} is a clopen halfspace in {0, 1}^{κ} and the collection {H_{α}}_{α<κ} is point-separating. Finally, the median of a, b, c ∈ {0, 1}^{κ} is the characteristic function of (a^{−1}(1) ∩ b^{−1}(1)) ∪ (a^{−1}(1) ∩ c^{−1}(1)) ∪ (b^{−1}(1) ∩ c^{−1}(1)). The following lemma is an analogue of the result of van Mill and Wattel [6].

Lemma 2.1 (cf. [7, Lemma I.3.16]). Every Boolean median space of weight ⩽ κ is isomorphic to a closed median-stable subset of a Cantor cube {0, 1}^{κ}.

Proof. Let X be a Boolean median space of weight ⩽ κ. Then there exists a subbase of the topology {H_{α}}_{α<κ} consisting of clopen halfspaces in X. Put j(x)(α) = 1 iff x ∈ H_{α}. This defines a map j: X → {0, 1}^{κ}, which is a topological embedding. By [7, Prop. I.1.12] it remains to show that j([a, b]) = [j(a), j(b)] ∩ j(X) for each a, b ∈ X. If x ∈ [a, b] then j(a)(α) = j(b)(α) = 1 implies j(x)(α) = 1 and j(x)(α) = 1 implies j(a)(α) = 1 or j(b)(α) = 1. This means that j(x) ∈ [j(a), j(b)]. If x ∉ [a, b] then taking y = m(a, b, x) we get x ≠ y and hence there exists an α < κ with y ∈ H_{α} and x ∉ H_{α}. Now a, b ∈ H_{α} and therefore j(a)(α) = j(b)(α) = 1 while j(x)(α) = 0. Hence j(x) ∉ [j(a), j(b)]. □

Theorem 2.2. Let X be a Boolean median space of weight κ ⩾ ω. There exists an inverse system of Boolean median spaces S = \{X_{α}, p_{α}^{β}, α < β < κ\} such that X = \underleftarrow{lim} S and

(1) |X_{0}| = 1.

(2) X_{λ} = \underleftarrow{lim}\{X_{α} : α < λ\}, for a limit ordinal λ < κ.

(3) For each α < κ there exist closed convex sets A_{α}, B_{α} ⊂ X_{α} such that A_{α} ∪B_{α} = X_{α}, X_{α+1} = (A_{α} × {0}) ∪ (B_{α} × {1}), p_{α}^{α+1} is the projection and the convexity of X_{α+1} is inherited from the product X_{α} × {0, 1}.

Proof. In view of Lemma 2.1 we may assume that X is a closed median-stable subspace of {0, 1}^{κ}. Set X_{α} = {x|α : x ∈ X}. Let p_{α}^{β}: X_{β} → X_{α} be the projection. Clearly, each p_{α}^{β} is continuous cp and p_{α}^{β}p_{β}^{γ} = p_{α}^{γ}. Let

A_{α} ={x|α : x(α) = 0, x ∈ X},
B_{α} ={x|α : x(α) = 1, x ∈ X}.

We have A_{α} ∪ B_{α} = X_{α} and A_{α}, B_{α} are closed. One can easily check that X = \underleftarrow{lim} S and the conditions (1), (2) are satisfied. Clearly X_{α+1} = (A_{α} × {0}) ∪ (B_{α} × {1}). It remains to show that A_{α}, B_{α} are convex. Let x|α, y|α ∈ A_{α} and z|α ∈ [x|α, y|α], where x, y, z ∈ X. Setting v = m(x, y, z) we see that v|α = z|α and v(α) = m(x(α), y(α), z(α)) = 0; hence z|α ∈ A_{α}. □

3. Main result

The following lemma is known, we present a proof for the sake of completeness.

Lemma 3.1. Let P be a closed under the complements subbase of a zero-dimensional compact topological space X and let B be the collection of all finite intersections of sets from P. Then every clopen subset of X can be partitioned into a finite number of members of B.

Proof. Set 𝔐 = {M_{1} ∪ ⋯ ∪ M_{n} : M_{i} ∈ B, M_{i} ∩ M_{j} = ∅ for i ≠ j, n ∈ ℕ}. Since B ⊂ 𝔐 and B is an open base of X, it is enough to show that 𝔐 is an algebra of sets. Fix B, C ∈ 𝔐 and let B and C have partitions M_{1} ∪ ⋯ ∪ M_{n} and N_{1} ∪ ⋯ ∪ N_{k} respectively, where M_{i}, N_{j} ∈ B. We have B ∩ C = ⋃_{i,j} M_{i} ∩ N_{j} and M_{i} ∩ N_{j}’s are pairwise disjoint. It follows that B∩C ∈ 𝔐. Now observe that X\M ∈ 𝔐 for M ∈ B. Indeed, if M = H_{1}∩⋯∩H_{n} where H_{i} ∈ P for i ⩽ n, then X \ M has a partition into sets of the form H_{1}^{ε(1)} ∩ ⋯ ∩ H_{n}^{ε(n)} where ε: {1, …, n} → {−1, 1} is a function not equal constantly to 1 and H_{i}^{1} = H_{i}, H_{i}^{−1} = X \H_{i}. Finally, if B = M_{1} ∪ ⋯ ∪ M_{n} ∈ 𝔐 where M_{i} ∈ B then the set X \ B = ⋂_{i⩽n}(X \ M_{i}) does belong to 𝔐. This completes the proof. □

Theorem 3.2. If X is a retract of a Cantor cube then there exists a convexity in X such that X is a Boolean median space.

Proof. According to [2] and [4, Thm. 2.7] we can represent X as the limit of an inverse system S = {X_{α}, p_{α}^{β}, α < β < τ} with the following properties:

(1) |X_{0}| = 1,

(2) X_{γ} = \underleftarrow{lim}\{X_{α}, α < γ\} for limit ordinals γ < τ,

(3) X_{α+1} = (X_{α} × {0}) ∪ (U_{α} × {1}) where U_{α} is clopen in X_{α} and p_{α}^{α+1} : X_{α+1} → X_{α} is the projection.

We define inductively suitable convexities in X_{α}’s in such a way that each X_{α} becomes a Boolean median space and each p_{α}^{α+1} becomes cp. Suppose that this is already done for all ξ < γ and assume that γ = α + 1. By Lemma 3.1, U_{α} = G_{1} ∪ ⋯ ∪ G_{n} where G_{i}’s are pairwise disjoint clopen and convex. Hence X_{α+1} = X_{α} ⊕G_{1} ⊕⋯⊕G_{n} and p_{α}^{α+1} is the superposition of n projections of the form X_{α} ⊕ G_{1} ⊕ ⋯ ⊕ G_{i+1} → X_{α} ⊕ G_{1} ⊕ ⋯ ⊕ G_{i}. Thus we may assume that U_{α} is convex. Now X_{α+1} is a median-stable subset (the union of two convex sets) of the product X_{α} × {0, 1}. It follows that X_{α+1} with the subspace convexity is a Boolean median space. Clearly p_{α}^{α+1} is cp. If γ is a limit ordinal and convexities G_{α} are already defined for α < γ then, by Proposition 1.2, X_{γ} with the convexity of the limit is a Boolean median space. This completes the proof. □

Remark 3.3. Actually, we have proved that if X is a zero-dimensional Dugundji space then there exists a convexity G in X and an inverse system of Boolean median spaces S = \{(X_{α}, G_{α}), p_{α}^{β}, α < β < τ\} such that (X, G) = \underleftarrow{lim} S and S has properties (1)−(3) above, with X_{α+1} = X_{α} ⊕ U_{α}, where U_{α} is clopen and convex in X_{α}. On the other hand, by Haydon’s Theorem [2], the inverse limit of such a system S is a topological zero-dimensional Dugundji space.

Theorem 3.4. Every retract of a Cantor cube has a binary subbase closed under the complements.

Proof. The desired subbase consists of the all clopen halfspaces, with respect to the convexity given by Theorem 3.2. □

The example below shows that the converse does not hold.

Example 3.5. Consider a one-point compactification ακ of the discrete space of cardinality κ. Let P be the collection of all one-element subsets of κ and all their complements. One can check that P is a binary subbase. On the other hand, if κ > ω then ακ is not dyadic and therefore cannot be a Dugundji space.

References

[1] J.R. Calder, Some elementary properties of interval convexities, J. London Math. Soc. (2),3 (1971) 422-428.

[2] R. Haydon, On a problem of Peczyski: Miljutin spaces, Dugundji spaces and AE(0-dim), Studia Math. 52 (1974) 23-31.

[3] L. Heindorf, Zero-dimensional Dugundji spaces admit profinite lattice structures, Comm. Math. Univ. Carol. 33, No. 2 (1992) 329-334.

[4] S. Koppelberg, Projective Boolean Algebras, Chapter 20 of: J.D. Monk (ed.), Handbook of Boolean Algebras, vol. 3, Amsterdam 1989.

[5] J. van Mill, Supercompactness and Wallman Spaces, Mathematisch Centrum, Amsterdam 1977.

[6] J. van Mill, E. Wattel, An external characterization of spaces which admit binary normal subbases, Amer. J. Math. 100 (1978) 987-994.

[7] M. van de Vel, Theory of Convex Structures, North-Holland, Amsterdam 1993.

Wiesław Kubiś, Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice

E-mail address: kubis@ux2.math.us.edu.pl

Andrzej Kucharski, Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice

E-mail address: akuchar@ux2.math.us.edu.pl