Abstract
We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.
Highlights
We introduce the topology on a set X as in Definition 7 by declaring each point of X \ X ∗ to be isolated and a basic open neighbourhood of a point ( p, k, ω ) ∈ X ∗ to be of the form {( p, k, ω )} ∪ {( p, k, n) : n ∈ F } for a given element F ∈ p
The topic of this paper is related to a long-standing open problem of van Douwen about the existence in ZFC alone of a countably compact group without non-trivial convergent sequences
(The existence of such a group in some additional set-theoretic axioms, such as Continuum Hypothesis (CH) or Martin’s Axiom (MA), is well-known.) it was noted in ([5], Example 5.7) that a solution to this problem would bring a positive solution to Question 1 (ii) and to the weaker Question 1 (i)
Summary
A space X is selectively pseudocompact (called strongly pseudocompact) provided that, for every sequence {Un : n ∈ N} of non-empty open subsets of X, one can choose a point xn ∈ Un for all n ∈ N in such a way that the resulting sequence { xn : n ∈ N} has a p-limit in X for some free ultrafilter p on. A space X is (i) strongly P-bounded provided that, for every sequence {Un : n ∈ N} of non-empty open subsets of X, one can choose a point xn ∈ Un for all n ∈ N in such a way that the resulting sequence { xn : n ∈ N} has a p-limit in X for every p ∈ P;. A space is pseudocompact if every real-valued continuous function on it is bounded
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