Article contents
SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL
Published online by Cambridge University Press: 07 August 2017
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
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.
Keywords
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 110 - 118
- Copyright
- © 2017 Australian Mathematical Publishing Association Inc.
References
Gabriyelyan, S. S. and Morris, S. A., ‘Free topological vector spaces’, Topology Appl.
223 (2017), 30–49.Google Scholar
Gabriyelyan, S. S. and Morris, S. A., ‘Embedding into free topological vector spaces on compact metrizable spaces’. Preprint.Google Scholar
Katz, E., Morris, S. A. and Nickolas, P., ‘A free subgroup of the free abelian topological group on the unit interval’, Bull. Lond. Math. Soc.
14 (1982), 399–402.Google Scholar
Mack, J., Morris, S. A. and Ordman, E. T., ‘Free topological groups and the projective dimension of a locally compact abelian groups’, Proc. Amer. Math. Soc.
40 (1973), 303–308.Google Scholar
Tkachenko, M. G., ‘On completeness of free abelian topological groups’, Soviet Math. Dokl.
27 (1983), 341–345.Google Scholar
You have
Access
- 1
- Cited by