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2017•Bulletin of the Australian Mathematical SocietyOpen access

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

Saak Gabriyelyan, Sidney A. Morris

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Abstract

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.

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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.

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Available abstract

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.

Key concepts: Mathematics, Subspace topology, Unit interval, Topological vector space, Tychonoff space, Linear subspace, Topological group, Interval (graph theory)

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