Domain invariance in infinite-dimensional linear spaces

Abstract:

Let $X$ be an infinite-dimensional locally convex linear space. It is known that $X$ is homeomorphic to a subspace of $X$ which is not open. We prove that every Banach space $B$ contains a dense linear subspace $L$ with the following property: If $U \subseteq L$ is open and if $f:U \to L$ is continuous and injective, then there exists a dense open $V \subseteq U$ such that (a) $\begin {gathered}f(V) \end {gathered}$ is open in $\begin {gathered} L \end {gathered}$, and (b) $f:V \to f(V)$ is a homeomorphism. As a consequence, $L$ and $L \times {\mathbf {R}}$ are not homeomorphic.