Partition-complete paracompact \(k\)-spaces are prescribed by closed maps (Q1304866)

A Čech-complete space is a partition-complete space, and a partition-complete space is a sieve-complete space. A sieve-complete space coincides with a complete metrizable space in a metrizable space. Partition-completeness and sieve-completeness are all preserved by open maps and perfect maps, and, more generally, by tri-quotient maps. The purpose of this paper is to study the preservation of partition-completeness and sieve-completeness by closed maps. The main results are that, let \(f:X\to Y\) be a closed map from a paracompact space \(X\) onto a space \(Y\), (1) if \(X\) is a partition-complete \(k\)-space, then \(Y\) is also a partition-complete space; (2) if \(X\) is a sieve-complete space, then \(Y\) is a sieve-complete space if and only if \(\partial f^{-1}(y)\) is a compact subset in \(X\) for every \(y\in Y\).