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EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 20 August 2019, pp. 311-324
- Print publication:
- April 2020
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- Article
- Export citation
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It is proved that the free topological vector space
$\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space 
$\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube 
$[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval 
$[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space 
$\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to 
$\mathbb{V}(X\oplus X)$, where 
$X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.
