On d-paracompact \(\sigma{}\)-spaces
This interesting paper contains some beautiful results concerning \(d\)- paracompact spaces which have remarkable consequences for developable spaces. Recall that a topological space \(X\) is called \(d\)-paracompact if for every open cover \({\mathcal U}\) of \(X\) there exists a developable \(T_ 1\)-space \(Y\), an open cover \({\mathcal V}\) of \(Y\), and a continuous mapping \(f\) from \(X\) onto \(Y\) such that \(f^{-1}({\mathcal V})\) refines \({\mathcal U}\). It is called a \(\sigma\)-space if it has a \(\sigma\)-closure-preserving closed network. Note that every \(\sigma\)-space is perfect, i.e. its closed sets are \(G_ \delta\)-sets. In the following all spaces are assumed to be \(T_ 1\). Theorem 1: Every subspace of a \(d\)-paracompact \(\sigma\)-space is a \(d\)-paracompact \(\sigma\)-space. Theorem 2: Every perfect image of a perfectly \(d\)-paracompact space (i.e. perfect and \(d\)- paracompact space) is perfectly \(d\)-paracompact. Theorem 3: Every perfect image of a \(d\)-paracompact \(\sigma\)-space is a \(d\)-paracompact \(\sigma\)- space. Whether \(d\)-paracompactness alone is preserved by perfect mappings remains an open problem (see Problem \(L\) in our survey [Diss. Math. 284 (1989; Zbl 0714.54030)]). Example 1: There exists a closed mapping from a \(d\)-paracompact \(\sigma\)-space onto a \(T_ 1\)-space which is not \(d\)- paracompact. Theorem 4: Every first countable closed image of a \(d\)- paracompact \(\sigma\)-space is a \(d\)-paracompact \(\sigma\)-space. Theorem 5: A closed image of a developable space is developable if and only if it is first countable. This result generalizes the classical Hanai-Morita- Stone-Theorem from metrizable spaces to developable spaces. Theorem 6: A \(T_ 1\)-space \(X\) is developable if and only if \(X \times Y\) is \(d\)- paracompact for every developable space \(Y\). Theorem 7: If \(X\) is a \(d\)- paracompact \(P\)-space and \(Y\) is a metacompact developable space, then \(X\times Y\) is \(d\)-paracompact. In particular, every product of a perfectly \(d\)-paracompact space and a metacompact developable space is \(d\)-paracompact. Example 2: There exists a semi-metrizable space which is not \(d\)-paracompact. This result settles Problem I of our above mentioned survey [loc. cit.].
scientific article; zbMATH DE number 61127