Products of sequential CLP-compact spaces are CLP-compact

It is shown that the product of finitely many sequential, CLP-compact spaces is CLPcompact. The class of spaces which have the property that every cover by clopen sets — sets which are both open and closed — has a finite subcover was introduced by A. Šostak in [1] under the name CBcompact. These spaces are now known as CLP-compact spaces and their study is linked to the question of whether the algebra of clopen sets in the product of two topological spaces is equal to the product of the corresponding algebras of clopen sets in the factors. The problem of finding weak hypotheses under which the product of CLP-compact spaces is CLP-compact should still be considered to be open even though some progress has been recorded. In Theorem 3.2 of [2] it is shown that the product of CLP-compact spaces of weight smaller than p is CLP-compact provided that the clopen algebras are countably generated. Question 6.1 of [2] asks whether the hypothesis on weight can be relaxed: For example, can the space simply be assumed to be first countable. It will be shown in this short note that the answer is positive and, in fact, much less suffices. However, several interesting questions remain and are listed at the end. Definition 0.1. If τ is a topology on the set X then let τ clp = τ ∩ {X \ U | U ∈ τ } be the set of all sets in τ which are clopen. Notation 0.1. If (X1, τ1) and (X2, τ2) are topological spaces then τ1 ⊗ τ2 will denote the product topology on X1 ×X2 — in other words, τ1 ⊗ τ2 is generated by the base {A×B | (A,B) ∈ τ1 × τ2}. Recall that a topological space is said to be sequential if any subset A of it is closed if and only if it contains the limit of every convergent sequence from A. Theorem 0.1. If (X1, σ1) and (X2, σ2) are zero-dimensional, compact, topological spaces and τ1 ⊇ σ1 and τ2 ⊇ σ2 are finer topologies on X1 and X2 such that • (X1, τ1) is sequential • and σ 1 = τ clp 1 • and σ 2 = τ clp 2 then (σ1 ⊗ σ2) = (τ1 ⊗ τ2). Proof. Let U ∈ (τ1× τ2). It will be shown that U ∈ (σ1×σ2). For any set Z ⊆ X1×X2 and x ∈ X1 let Zx = {y ∈ X2 | (x, y) ∈ Z }. For any A ⊆ X2 let A∗ = {x ∈ X1 | Ux = A}. It will first be shown that if A ∈ σ 2 then A∗ ∈ τ clp 1 . To see that X1 \ A∗ ∈ τ1 suppose that x ∈ X1 \ A∗ and x belongs to the τ1-closure of A∗. Let y ∈ A4Ux. Then (x, y) belongs to the τ1⊗ τ2-closure of A∗×{y}. Moreover, A∗×{y} ⊆ U if (x, y) / ∈ U and A∗ × {y} ∩ U = ∅ if (x, y) ∈ U contradicting that U ∈ (τ1 × τ2). To see that A∗ ∈ τ1 suppose not and use the hypothesis that (X1, τ1) is sequential to choose a sequence {xn}n=0 ⊆ X1 \A∗ such that limn→∞ xn = x ∈ A∗. In other words, Uxn 6= A for each n yet Ux = A. By restricting to a subsequence it is possible to assume that one of the following holds: (0.1) (∀n ∈ N) Uxn \ A 6= ∅ Research for this paper was partially supported by NSERC of Canada. 1