A characterization of free locally convex spaces over metrizable spaces which have countable tightness (Q2815614)

For a topological space \(X\), the free locally convex space (l.c.s.) \(L(X)\) is an l.c.s. together with a topological embedding \(i:X\hookrightarrow L(X)\) such that every continuous \(f:X\to E\) with values in an l.c.s. \(E\) induces a unique continuous linear \(\bar f: L(X) \to E\) with \(\bar f \circ i=f\) (it is the free vector space over \(X\) endowed with a suitable locally convex topology).NEWLINENEWLINEA topological space is said to have countable tightness if the closure of every subset coincides with the union of the closures of all countable subsets. Besides metrizable spaces, strong duals of Fréchet-Schwartz spaces have this property.NEWLINENEWLINEThe main result of this article characterizes for a metric space \(X\) that \(L(X)\) has countable tightness (and is then even a so-called Pytkeev \(\aleph_0\)-space) if and only if \(X\) is separable.