Products and h-homogeneity

arXiv:0911.1023v2 [math.GN] 4 Dec 2011 PRODUCTS AND H-HOMOGENEITY ANDREA MEDINI Abstract. Building on work of Terada, we prove that h-homogeneity is pro- ductive in the class of zero-dimensional spaces. Then, by generalizing a re- sult of Motorov, we show that for every non-empty zero-dimensional space X there exists a non-empty zero-dimensional space Y such that X × Y is h-homogeneous. Also, we simultaneously generalize results of Motorov and Terada by showing that if X is a space such that the isolated points are dense then Xκ is h-homogeneous for every infinite cardinal κ. Finally, we show that a question of Terada (whether Xω is h-homogeneous for every zero-dimensional first-countable X) is equivalent to a question of Motorov (whether such an infinite power is always divisible by 2) and give some partial answers. All spaces in this paper are assumed to be Tychonoff. It is easy to see that every zero-dimensional space is Tychonoff. For all undefined topological notions, see [7]. For all undefined Boolean algebraic notions, see [9]. Recall that a subset of a space is clopen if it is closed and open. Definition 1. A space X is h-homogeneous (or strongly homogeneous) if all non- empty clopen subsets of X are homeomorphic to each other. The Cantor set, the rationals and the irrationals are examples of h-homogeneous spaces. Every connected space is h-homogeneous. A finite space is h-homogeneous if and only if it has size at most 1. The concept of h-homogeneity has been studied (mostly in the zero-dimensional case) by several authors: see [10] for an extensive list of references. We will denote by Clop(X) the Boolean algebra of the clopen subsets of X. Recall that a Boolean algebra A is homogeneous if A ↾a is isomorphic to A for every non-zero a ∈A, where A ↾a denotes the relative algebra {x ∈A : x ≤a}. If X is h-homogeneous then Clop(X) is homogeneous; the converse holds if X is compact and zero-dimensional (see the remarks following Definition 9.12 in [9]). 1. The productivity of h-homogeneity In [20], the productivity of h-homogeneity is stated as an open problem (see also [10] and [11]), and it is shown that the product of zero-dimensional h-homogeneous spaces is h-homogeneous provided it is compact or non-pseudocompact (see The- orem 3.3 in [20]). The following theorem, proved by Terada under the additional assumption that X is zero-dimensional (see Theorem 2.4 in [20]), is the key ingre- dient in the proof. Recall that a collection B consisting of non-empty open subsets of a space X is a π-base if for every non-empty open subset U of X there exists V ∈B such that V ⊆U. Theorem 2 (Terada). Assume that X is non-pseudocompact. If X has a π-base consisting of clopen sets that are homeomorphic to X then X is h-homogeneous. Date: July 13, 2010. 1 2 ANDREA MEDINI The proof of Theorem 2 uses the fact that a zero-dimensional non-pseudocompact space can be written as the disjoint union of infinitely many of its non-empty clopen subsets (the converse is also true, trivially). However, that is the only consequence of zero-dimensionality that is actually used (see Appendix A). Therefore such as- sumption is redundant by the following lemma, whose proof we leave to the reader. Lemma 3. Assume that X is non-pseudocompact. If X has a π-base consisting of clopen sets then X can be written as the disjoint union of infinitely many of its non-empty clopen subsets. Using Theorem 2 one can easily prove the following. Theorem 4 (Terada). Assume that X = Q i∈I Xi is non-pseudocompact. If Xi is h-homogeneous and it has a π-base consisting of clopen sets for every i ∈I then X is h-homogeneous. For proofs of the following basic facts about βX, see Section 11 in [12]. Given any open subset U of X, define Ex(U) = βX \ clβX(X \ U). It is easy to see that Ex(U) is the biggest open subset of βX such that its intersection with X is U. If C is a clopen subset of X then Ex(C) = clβX(C), hence Ex(C) is clopen in βX. Furthermore, the collection {Ex(U) : U is open in X} is a base for βX. Remark. It is not true that βX is zero-dimensional whenever X is zero-dimensional (see Example 6.2.20 in [7] or Example 3.39 in [22]). If βX is zero-dimensional then X is called strongly zero-dimensional. We will need the following theorem (see Theorem 8.25 in [22]); see also Exercise 3.12.20(d) in [7]. Recall that a subspace Y of X is C∗-embedded in X if every bounded continuous function f : Y −→R admits a continuous extension to X. Theorem 5 (Glicksberg). Assume that X = Q i∈I Xi is pseudocompact. Then X is C∗-embedded in Q i∈I βXi. Remark. The reverse implication is also true, under the additional assumption that Q j̸=i Xj is infinite for every i ∈I. Such assumption is clearly not needed in the above statement (see Proposition 8.2 in [22]). Proposition 6. Assume that X × Y is pseudocompact. If C is a clopen subset of X × Y then C can be written as the union of finitely many open rectangles. Proof. It follows from Theorem 5 that X × Y is C∗-embedded in βX × βY . By the

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