The closed images of metric spaces and Fréchet \(\aleph\)-spaces. (English) Zbl 0643.54035

We recall that a cover \({\mathcal P}\) of a space X is called a k-network if, whenever \(K\subset U\) with K compact and U open in X, then \(K\subset \cup {\mathcal P}'\subset U\) for some finite \({\mathcal P}'\subset {\mathcal P}\). A space X is called an \(\aleph\)-space if it has a \(\sigma\)-locally finite k- network. The author uses the notion of k-network to show the three following statements and to characterize k-semi-stratifiable spaces defined by D. J. Lutzer [Gen. Topol. Appl. 1, 43-48 (1971; Zbl 0211.257)] by means of g-functions. (1) Let \(f: X\to Y\) be a closed map with X metric. Then Y is an \(\aleph\)-space if and only if every \(\partial f^{-1}(y)\) is Lindelöf. (2) (CH). A Lashnev space (i.e., closed image of a metric space) X is a Fréchet \(\aleph\)-space if and only if the character of X is less than or equal to \(2^{\omega}\). (3) Every regular space X with a \(\sigma\)-closure preserving point-countable k-network is metrizable if X is a countably bi-k-space in the sense of E. Michael [Gen. Topol. Appl. 2, 91-138 (1972; Zbl 0238.54009)], for example.
Reviewer: Y.Tanaka


54E20 Stratifiable spaces, cosmic spaces, etc.
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54E35 Metric spaces, metrizability
54D55 Sequential spaces