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SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

Part of: Topological linear spaces and related structures Fairly general properties Generalities

Published online by Cambridge University Press:  07 August 2017

SAAK S. GABRIYELYAN  and
SIDNEY A. MORRIS Open the ORCID record for SIDNEY A. MORRIS [Opens in a new window]
SAAK S. GABRIYELYAN
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva PO 653, Israel email [email protected]
SIDNEY A. MORRIS*
Affiliation:
Faculty of Science and Technology, Federation University Australia, PO Box 663, Ballarat, Victoria 3353, Australia Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia email [email protected]
*
✉[email protected]
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Abstract

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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.

Keywords

free topological vector spacefree topological groupfree abelian topological groupembeddinglocally compact

MSC classification

Primary: 46A03: General theory of locally convex spaces
Secondary: 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D50: $k$-spaces
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

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