Strongly $σ$-metrizable spaces are super $σ$-metrizable

A topological space $X$ is called strongly $σ$-metrizable if $X=\bigcup_{n\inω}X_n$ for an increasing sequence $(X_n)_{n\inω}$ 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 $X_n$, $n\inω$, then $X$ is called super $σ$-metrizable. Answering a question of V.K.Maslyuchenko and O.I.Filipchuk, we prove that a topological space is strongly $σ$-metrizable if and only if it is super $σ$-metrizable.

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