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The impact of the Bohr topology on selective pseudocompactness
- Source :
- Topology and its Applications. 264:498-506
- Publication Year :
- 2019
- Publisher :
- Elsevier BV, 2019.
-
Abstract
- Recall that a space X is selectively pseudocompact if for every sequence { U n : n ∈ N } of non-empty open subsets of X one can choose a point x n ∈ U n for all n ∈ N such that the resulting sequence { x n : n ∈ N } has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by Garcia-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first listed author. In 2015, Garcia-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.
- Subjects :
- Sequence
Group (mathematics)
010102 general mathematics
General Topology (math.GN)
Mathematics::General Topology
Space (mathematics)
Topology
01 natural sciences
Bohr model
010101 applied mathematics
Mathematics::Group Theory
symbols.namesake
Limit point
FOS: Mathematics
symbols
Countable set
Primary: 22A05, Secondary: 22D35, 54D30
Geometry and Topology
Topological group
0101 mathematics
Topology (chemistry)
Mathematics - General Topology
Mathematics
Subjects
Details
- ISSN :
- 01668641
- Volume :
- 264
- Database :
- OpenAIRE
- Journal :
- Topology and its Applications
- Accession number :
- edsair.doi.dedup.....683280b9bcac82b3331aa06842ba0723