On strong sequence-covering compact mappings. (English) Zbl 0927.54030

The purpose of this paper is a discussion about strong sequence-covering (compact-covering) and compact images of metric spaces. Let \(f:X\to Y\) be a continuous and onto map. \(f\) is a compact map if each \(f^{-1}(y)\) is compact; \(f\) is a compact-covering map if each compact subset of \(Y\) is the image of some compact subset of \(X\); \(f\) is a strong sequence-covering map if each convergent sequence of \(Y\) is the image of some convergent sequence of \(X\). In this paper the main results are that for a \(T_2\)-space \(X\)
(1) \(X\) is a strong sequence-covering and compact image of a metric space if and only if \(X\) has a sequence of point-finite cs-covers such that \(\{st(x,{\mathcal U}_n): n\in\mathbb{N}\}\) forms a network of \(x\) for each \(x\in X\);
(2) \(X\) is a strong sequence-covering, compact-covering and compact image of a metric space if and only if \(X\) has a weak development consisting of point-finite cs-covers.
The strong sequence-covering maps were called sequence-covering maps by F. Siwiec in [General Topol. Appl. 1, 143-154 (1971; Zbl 0218.54016)] and sequence-covering maps were renamed another kind of maps by G. Gruenhage, E. Michael and Y. Tanaka in [Pac. J. Math. 113, 303-332 (1984; Zbl 0561.54016)].
Reviewer: Shou Lin (Fujian)


54E99 Topological spaces with richer structures
54C10 Special maps on topological spaces (open, closed, perfect, etc.)