Determining a weakly locally compact group topology by its system of closed subgroups. (English) Zbl 0921.22001
Andima, Susan (ed.) et al., Papers on general topology and applications. Papers from the 8th summer conference at Queens College, New York, NY, USA, June 18–20, 1992. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 728, 248-261 (1994).
The authors show that a topological group \(G\) is locally pseudocompact iff \(G\) is locally bounded and, for every closed \(G_\delta\)-subgroup \(H\) of \(G\), the quotient space \(G/H\) is locally compact. This result is applied to give a partial positive answer to a problem posed by K. A. Ross in 1965: If \({\mathcal T}_1\) and \({\mathcal T}_2\) are locally pseudocompact group topologies on an Abelian group \(G\) such that \((G,{\mathcal T}_1)\) and \((G,{\mathcal T}_2)\) have the same closed subgroups, then \({\mathcal T}_1\subseteq {\mathcal T}_2\) implies \({\mathcal T}_1 = {\mathcal T}_2\). The same conclusion remains valid for certain non-Abelian groups, for example, if \((G,{\mathcal T}_1)\) is locally compact and \((G,{\mathcal T}_2)\) is locally pseudocompact.
For the entire collection see [Zbl 0903.00047].
For the entire collection see [Zbl 0903.00047].
Reviewer: Michael G.Tkachenko (México)
MSC:
22A05 | Structure of general topological groups |
54H11 | Topological groups (topological aspects) |
22D05 | General properties and structure of locally compact groups |