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On \(k_r\)-spaces and \(k\)-spaces. (English) Zbl 0996.54037

From the text: The following question was raised by S. Lin in [Acta Math. Sin. 34, No. 1, 7-11 (1991; Zbl 0760.54009)]. If a regular \(k_r\)-space \(X\) has a locally countable \(k\)-network consisting of compact subsets, is then \(X\) a \(k\)-space?
If the answer of this question is positive then quotient \(ss\)-mappings and \(R\)-quotient \(ss\)-mappings on locally compact metrizable spaces are equivalent. In this note, we give a necessary and sufficient condition for a regular \(k_r\)-space \(X\) with a locally countable closed \(k\)-network to be a \(k\)-space. Our result shows that even if the condition “compact subsets” of this question is replaced by the weaker condition “closed \(k\)-subspaces”, this question can be answered positively.

MSC:

54D50 \(k\)-spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0760.54009
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