## J. Gerlits, I. Juh\'asz, Z. Szentmikl\'ossy

*Two improvements on Tka\v cenko's addition theorem *

Comment.Math.Univ.Carolinae 46,4 (2005) 705-710. **Abstract:**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 $N(\Bbb R)$ = $cov (\Cal M)$ left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tka\v cenko from 1979; (A) also answers a question that was raised by Arhangel'ski\v {i} and improves a result of Gruenhage.

**Keywords:** $D$-space, left separated, compact, countably compact, scattered space, Nov\'ak number

**AMS Subject Classification:** 54D55, 54G12, 54A25

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