Precalibers of \(\sigma\)-compact topological groups
type: Item
instance of: scholarly article
title: Precalibers of \(\sigma\)-compact topological groups
zbMATH Open document ID: 0605.22003
DOI: 10.1007/BF01157033
author: B. George
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: In 1981 the reviewer had shown that each \(\sigma\)-compact topological group G has a countable cellularity, but the cardinal \(\aleph_ 1\) (and any other) need not be a caliber of G. This is the motivation of A. V. Arkhangel'skij's question if the cardinal \(\aleph_ 1\) is a precaliber of each \(\sigma\)-compact topological group. Obviously, \(MA+\rceil CH\) implies that the answer is ''yes''. In the paper it is shown that if a model \({\mathfrak N}\) of ZFC is obtained by adding of \(\aleph_ 1\) Cohen reals to some basic model \({\mathfrak M}\), then \({\mathfrak N}\vDash\) ''there exists a \(\sigma\)-compact topological group \(G^*\) such that \(\aleph_ 1\) is not a precaliber of \(G^*.''\) Moreover, one can take \(G^*=F(X)\) or \(G^*=A(X)\), where F(X) (A(X)) is the free (abelian) topological group over the Alexandrov compactification X of a discrete set of cardinality \(\aleph_ 1.\) A note. It seems to be interesting that any regular cardinal \(\tau \geq \aleph_ 2\) is a precaliber of each Lindelöf topological group.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 3977294
zbMATH Keywords: generic extension
zbMATH Keywords: \(\sigma \)-compact topological group
zbMATH Keywords: cellularity
zbMATH Keywords: precaliber
zbMATH Keywords: Cohen reals
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
cites work: Q5840188
cites work: Set theory. An introduction to independence proofs
cites work: Q4755386
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cites work: Q5547266
cites work: Q3734742
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description: scientific article; zbMATH DE number 3977294