## 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

### MSC:

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

### Citations:

Zbl 0211.257; Zbl 0238.54009