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On covering properties by regular closed sets. (English) Zbl 0893.54018
Summary: A topological space $$X$$ is [countably] rc-compact (rc-Lindelöf) if every [countable] cover of $$X$$ by regular closed sets has a finite (countable) subcover. It is established, among other results, that 1. A space is rc-compact iff its semiregularization is an extension of a compact extremally disconnected space; 2. An uncountable $$T_3$$ first countable crowded space is rc-Lindelöf iff it is a Luzin space, and 3. A countably rc-compact $$T_3$$ first countable or generalized ordered space is finite.

MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54G05 Extremally disconnected spaces, $$F$$-spaces, etc.
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