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Keywords:
$D$-space; left separated; compact; countably compact; scattered space; Novák number
$D$-space; left separated; compact; countably compact; scattered space; Novák number
Summary:
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.
References:
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