Summary: In this note, we study a metrizability problem of a subset \(A\) of a space \(X\). By the usual method of proving a space to be metrizable in general topology, we get the following conclusions: we show that if a set \(A\) is a bounded subset of a regular space \(X\) and \(X\) has a regular \(G_\delta\)-diagonal related to the set \(A\), then \(\overline{A}\) is metrizable; and we get that if \(F\) is a bounded strong zero-set of a regular space \(X\) and \(X\) has a regular \(G_\delta\)-diagonal related to the set \(F\), then \(F\) is a compact metrizable subspace of \(X\).