1 results
NOTES ON
$K$-TOPOLOGICAL GROUPS AND HOMEOMORPHISMS OF TOPOLOGICAL GROUPS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 87 / Issue 3 / June 2013
- Published online by Cambridge University Press:
- 30 August 2012, pp. 493-502
- Print publication:
- June 2013
-
- Article
-
- You have access
- Export citation
-
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.
