Strongly $\sigma$-metrizable spaces are super $\sigma$-metrizable

Abstract

A topological space $X$ is called strongly $\sigma$-metrizable if $X=\bigcup_{n\in\omega}X_n$ for an increasing sequence $(X_n)_{n\in\omega}$ of closed metrizable subspaces such that every convergence sequence in $X$ is contained in some $X_n$. If, in addition, every compact subset of $X$ is contained in some $X_n$, $n\in\omega$, then $X$ is called super $\sigma$-metrizable. Answering a question of V.K.Maslyuchenko and O.I.Filipchuk, we prove that a topological space is strongly $\sigma$-metrizable if and only if it is super $\sigma$-metrizable.