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|Free Locally Convex Spaces and the $k$-space Property|
Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$.
Keywords:free locally convex space, $k$-space, countable tightness
Categories:46A03, 54D50, 54A25