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Topological groups all continuous automorphisms of which are open. (English) Zbl 1437.22001

All topological groups in the paper under this review are Hausdorff. The notion of reversible topological space was introduced by M. Rajagopalan and A. Wilansky [J. Aust. Math. Soc. 6, 129–138 (1966; Zbl 0151.29602)]. A topological space \(X\) is called reversible if each continuous bijection of \(X\) onto itself is open (or equivalently, a homeomorphism). The authors introduce an analogue notion for topological groups (some sort of an “open mapping property”). A topological group \(G\) is called \(g\)-reversible if each continuous automorphism of \(G\) is open. They prove that each reversible topological group is \(g\)-reversible and many classical groups (Polish groups, \(\sigma\)-compact locally compact groups, minimal groups, abelian groups with the Bohr topology, connected locally compact groups) are \(g\)-reversible. Every subgroup of \(\mathbb{R}^n\) (\(n\in \mathbb{N}\)) is \(g\)-reversible. The authors give numerous examples of non-g-reversible groups. Many open problems are contained throughout the paper (for example: “describe \(g\)-reversible locally compact (abelian) groups”; “must a closed subgroup of a \(g\)-reversible abelian group \(G\) be \(g\)-reversible ?”; “must every closed subgroup of a \(g\)-reversible locally compact abelian group \(G\) be \(g\)-reversible?” etc). The authors introduce and to research the notion of hereditarily \(g\)-reversible. A topological group \(G\) is called hereditarily \(g\)-reversible if every subgroup of \(G\) is \(g\)-reversible (in the subspace topology). Some properties of \(g\)-reversibility of topological products are explored.

MSC:

22A05 Structure of general topological groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
22B05 General properties and structure of LCA groups
22D05 General properties and structure of locally compact groups
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54D45 Local compactness, \(\sigma\)-compactness
54H11 Topological groups (topological aspects)

Citations:

Zbl 0151.29602
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Full Text: DOI arXiv

References:

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