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Certain \(s\)-images of locally separable metric spaces. (English) Zbl 0858.54030

The authors consider characterizations of images of locally separable metric spaces under special compact-covering (or cs-covering) mapping, in terms of special \(k\)-networks (or cs-networks). The term special means locally countable, star-countable or point-countable, and a mapping of a locally separable metric space is said to be locally countable (star-countable or point-countable) if the image of a (equivalently, any) star-countable base of the domain is a locally countable (star-countable or point-countable) collection. In particular, point-countable mappings are the mappings with separable fibers (= s-mappings) defined on locally separable metric spaces.
For locally compact metric spaces the images can, easily, be characterized as spaces with the corresponding network consisting of compact metric sets. However, in the locally separable case, not all of the possible six combinations lead to answers.

MSC:

54E99 Topological spaces with richer structures
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