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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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