EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES
2019 ◽
Vol 101
(2)
◽
pp. 311-324
Keyword(s):
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.
Keyword(s):
2017 ◽
Vol 103
(3)
◽
pp. 402-419
◽
2017 ◽
Vol 97
(1)
◽
pp. 110-118
◽
1987 ◽
Vol 43
(2)
◽
pp. 224-230