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Axioms 2018, 7(4), 86; https://doi.org/10.3390/axioms7040086

Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

1
Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
2
Doctor’s Course, Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
*
Author to whom correspondence should be addressed.
This article is dedicated to Professor Alexander V. Arhangel’ski˘ı on the occasion of his 80th birthday.
Received: 31 July 2018 / Revised: 21 October 2018 / Accepted: 5 November 2018 / Published: 16 November 2018
(This article belongs to the Collection Topological Groups)
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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. View Full-Text
Keywords: pseudocompact; strongly pseudocompact; p-compact; selectively sequentially pseudocompact; pseudo-ω-bounded; non-trivial convergent sequence; separable; free precompact Boolean group; reflexive group; maximal space; ultrafilter space pseudocompact; strongly pseudocompact; p-compact; selectively sequentially pseudocompact; pseudo-ω-bounded; non-trivial convergent sequence; separable; free precompact Boolean group; reflexive group; maximal space; ultrafilter space
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Shakhmatov, D.; Yañez, V.H. Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets. Axioms 2018, 7, 86.

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