On the widest class of completeness-preserving covering maps (Q2074377)

The author [Topology Appl. 201, 269--273 (2016; Zbl 1337.54020)] sought for the widest class \(\mathfrak{H}\) of maps \(f: X\to Y\) between separable metrizable spaces that satisfies the following conditions: (1)\, Maps \(f\) from \(\mathfrak{H}\) are determined by their behaviour on some countable compact sets from \(Y\); (2)\, If \(X\) is a Polish space, then \(Y\) is also a Polish space; (3)\, Suppose \(f: X\to Y\) and \(g: Y\to Z\) are maps. If \(f\) and \(g\) belong to \(\mathfrak{H}\), then so does \(g\circ f\). In this paper \(H\)-covering maps are introduced and it is proposed that the class of \(H\)-covering continuous maps is the class \(\mathfrak{H}\). Finally, the author analyzes the following three kinds of maps in separable metric spaces: \(H\)-covering maps, \(s\)-covering maps and tri-quotient maps.