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Addition theorems and \(D\)-spaces. (English) Zbl 1090.54017
Summary: It is proved that if a regular space \(X\) is the union of a finite family of metrizable subspaces then \(X\) is a \(D\)-space in the sense of E. van Douwen. It follows that if a regular space \(X\) of countable extent is the union of a finite collection of metrizable subspaces then \(X\) is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a \(D\)-space. Some other new results on the properties of spaces which are unions of a finite collection of nice subspaces are obtained.

MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54F99 Special properties of topological spaces
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