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SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 07 August 2017, pp. 110-118
- Print publication:
- February 2018
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- Article
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- You have access
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For a Tychonoff space
$X$, let 
$\mathbb{V}(X)$ be the free topological vector space over 
$X$, 
$A(X)$ the free abelian topological group over 
$X$ and 
$\mathbb{I}$ the unit interval with its usual topology. It is proved here that if 
$X$ is a subspace of 
$\mathbb{I}$, then the following are equivalent: 
$\mathbb{V}(X)$ can be embedded in 
$\mathbb{V}(\mathbb{I})$ as a topological vector subspace; 
$A(X)$ can be embedded in 
$A(\mathbb{I})$ as a topological subgroup; 
$X$ is locally compact.
VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 78 / Issue 3 / December 2008
- Published online by Cambridge University Press:
- 01 December 2008, pp. 487-495
- Print publication:
- December 2008
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- Article
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- You have access
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