Geodesic
Two improvements on Tkačenko's addition theorem
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 705-710.

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We prove that (A) if a countably compact space is the union of countably many $D$ subspaces then it is compact; (B) if a compact $T_2$ space is the union of fewer than $N(\Bbb R)$ = $\operatorname{cov} (\Cal M)$ left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tkačenko from 1979; (A) also answers a question that was raised by Arhangel'ski\v{i} and improves a result of Gruenhage.
Classification : 54A25, 54D55, 54G12
Keywords: $D$-space; left separated; compact; countably compact; scattered space; Novák number 
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Gerlits, J.; Juhász, I.; Szentmiklóssy, Z. Two improvements on Tkačenko's addition theorem. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 705-710. http://geodesic.mathdoc.fr/item/CMUC_2005__46_4_a8/