## A.V. Arhangel'skii, R.Z. Buzyakova

*Addition theorems and $D$-spaces *

Comment.Math.Univ.Carolinae 43,4 (2002) 653-663. **Abstract:**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 Lindel\"of. 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.

**Keywords:** $D$-space, point-countable base, extent, metrizable space, Lindel\"of space

**AMS Subject Classification:** 54D20, 54F99

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