2 results
Free Locally Convex Spaces and the k-space Property
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 4 / 01 December 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 803-809
- Print publication:
- 01 December 2014
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- Article
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Let
$L\left( X \right)$ be the free locally convex space over a Tychonoff space 
$X$. Then $L\left( X \right)$ is a 
$k$-space if and only if $X$ is a countable discrete space. We prove also that 
$L\left( D \right)$ has uncountable tightness for every uncountable discrete space 
$D$.
A NOTE ON
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$-SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 1 / August 2014
- Published online by Cambridge University Press:
- 12 May 2014, pp. 144-148
- Print publication:
- August 2014
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In this paper, it is shown that every compact Hausdorff
$K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group 
$G$ that is a 
$K$-space is first countable if and only if 
$G$ is of point-countable type. It is proved that if a topological group 
$G$ is a 
$K$-space and has a locally paracompact remainder in some Hausdorff compactification, then 
$G$ is metrisable.
