Quasi-metrizability (Q1295363)

The author shows that quasi-(pseudo)metrizability of a topological space \(X\) is equivalent to the availability on \(X\) of a decreasing neighborhood base \((g(n,x))_{n\in \omega}\) at every \(x\in X\) such that for every countable and relatively locally finite \(A\subseteq X\) and \(n\in \omega\) we have \(g(\nu,g(\nu,A))\subseteq g(n,A)\) for some \(\nu\in \omega\) (dependent on \(A\) and \(n\)). The result is inspired by older metrization results of the author and known characterizations of \(\gamma\)-spaces. The proof is based on a result due to \textit{R. Fox} and the reviewer [Arch. Math. 41, 57-63 (1983; Zbl 0504.54028)].