The $k$-property and countable tightness of free topological vector spaces

Fucai Lin

Published 2017-06-07Version 1

The free topological vector space $V(X)$ over a Tychonoff space $X$ is a pair consisting of a topological vector space $V(X)$ and a continuous map $i=i_{X}: X\rightarrow V(X)$ such that every continuous mapping $f$ from $X$ to a topological vector space $E$ gives rise to a unique continuous linear operator $\overline{f}: V(X)\rightarrow E$ with $f=\overline{f}\circ i$. In this paper the $k$-property and countable tightness of free topological vector space $V(X)$ over a metrizable space $X$ are studied. For a metrizable space $X$, it is proved that the free topological vector space $V(X)$ is a $k$-space if and only if $X$ is a $k_{\omega}$-space, and the tightness of $V(X)$ is countable if and only $X$ is separable.