Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets†
Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
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
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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
of natural numbers and every sequence
of non-empty open subsets of G
, one can choose a point
in such a way that the resulting sequence
has a p
-limit in G
; that is,
for every neighbourhood V
. 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-
-bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum
, where each space
is either maximal or discrete, contains no infinite separable pseudocompact subsets.
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MDPI and ACS Style
Shakhmatov, D.; Yañez, V.H. Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets. Axioms 2018, 7, 86.
Shakhmatov D, Yañez VH. Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets. Axioms. 2018; 7(4):86.
Shakhmatov, Dmitri; Yañez, Víctor H. 2018. "Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets." Axioms 7, no. 4: 86.
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