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Keywords = free precompact Boolean group

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23 pages, 416 KB  
Article
Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets
by Dmitri Shakhmatov and Víctor Hugo Yañez
Axioms 2018, 7(4), 86; https://doi.org/10.3390/axioms7040086 - 16 Nov 2018
Cited by 3 | Viewed by 3798
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 [...] Read more.
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. Full article
(This article belongs to the Collection Topological Groups)
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