Strongly paracompact metrizable spaces (Q2804286)

A space \(Z\) is non-Archimedean if it has a base such that if \(B_1\) and \(B_2\) are members of this base with \(B_1\cap B_2\neq\emptyset\), then either \(B_1\subset B_2\) or \(B_2\subset B_1\). It was obtained by Yu. M. Smirnov that every strongly paracompact metrizable space can be mapped continuously onto a non-Archimedean metrizable space by an \(S\)-map. Strongly paracompact metrizable spaces are characterized in terms of special \(S\)-maps onto non-Archimedean metrizable spaces. A metrizable space is called strongly metrizable if it has a base which is the union of countably many star-finite open covers. A similar characterization of strongly metrizable spaces is obtained as well. The approach is based on a sieve-construction of ``metric''-continuous pseudo-sections of lower semicontinuous mappings.