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The free topological vector space $V(X)$ over a Tychonoff space $X$ is a pairconsisting 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 atopological 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 paperthe $k$-property and countable tightness of free topological vector space oversome generalized metric spaces are studied. The characterization of a space $X$is given such that the free topological vector space $V(X)$ is a $k$-space orthe tightness of $V(X)$ is countable. Furthermore, the characterization of aspace $X$ is also provided such that if the fourth level of $V(X)$ has the$k$-property or is of the countable tightness then $V(X)$ is too.

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