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Keywords:
$D$-space; point-countable base; extent; metrizable space; Lindelöf space
$D$-space; point-countable base; extent; metrizable space; Lindelöf space
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.
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