Geodesic
Addition theorems and $D$-spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 4, pp. 653-663

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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.
Classification : 54D20, 54E35, 54F99
Keywords: $D$-space; point-countable base; extent; metrizable space; Lindelöf space 
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Arhangel'skii, A. V.; Buzyakova, R. Z. Addition theorems and $D$-spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 4, pp. 653-663. http://geodesic.mathdoc.fr/item/CMUC_2002__43_4_a5/