Bounded tightness for locally convex spaces and spaces \(C(X)\) (Q1883355)

A locally convex space \(X\) is said to have the property of \textit{bounded tightness} if for every set \(A \subset X\) and every \(x \in \overline{A}\) there exists a bounded set \(B \subset A\) such that \(x \in \overline{B}\). \textit{B. Cascales} and \textit{J. Orihuela} [Math. Z. 195, 365--381 (1987; Zbl 0604.46011)] introduced a class, called class \({\mathcal G}\), of locally convex spaces containing, among other examples, metrizable spaces, (LF)-spaces, (DF)-spaces, and the space of distributions \({\mathcal D}'(\Omega)\). The main result of this interesting paper is that for spaces in the class \(\mathcal G\), metrizability is equivalent to the property of bounded tightness for the weak topology. Among other consequences, it is obtained that a metric space \(T\) is separable if the space \(C_p(T)\) has bounded tightness.