Precalibers of \(\sigma\)-compact topological groups
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.
scientific article; zbMATH DE number 3977294
scientific article