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- Title
NOTES ON $K$-TOPOLOGICAL GROUPS AND HOMEOMORPHISMS OF TOPOLOGICAL GROUPS.
- Authors
WANG, HANFENG; HE, WEI
- Abstract
In this paper, it is shown that there exists a connected topological group which is not homeomorphic to any $\omega $-narrow topological group, and also that there exists a zero-dimensional topological group $G$ with neutral element $e$ such that the subspace $X = G\setminus \{e\}$ is not homeomorphic to any topological group. These two results give negative answers to two open problems in Arhangel’skii and Tkachenko [Topological Groups and Related Structures (Atlantis Press, Amsterdam, 2008)]. We show that if a compact topological group is a $K$-space, then it is metrisable. This result gives an affirmative answer to a question posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl. 104 (2000), 181–190] in the category of topological groups. We also prove that a regular $K$-space $X$ is a weakly Fréchet–Urysohn space if and only if $X$has countable tightness.
- Subjects
TOPOLOGICAL groups; HOMEOMORPHISMS; SUBSPACES (Mathematics); URYSOHN equation; ARBITRARY constants; FRECHET spaces; MATHEMATICAL continuum
- Publication
Bulletin of the Australian Mathematical Society, 2013, Vol 87, Issue 3, p493
- ISSN
0004-9727
- Publication type
Article
- DOI
10.1017/S0004972712000561