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Countably \(I\)-compact spaces. (English) Zbl 0993.54022

Summary: We introduce the class of countably \(I\)-compact spaces as a proper subclass of countably \(S\)-closed spaces. A topological space \((X,T)\) is called countably \(I\)-compact if every countable cover of \(X\) by regular closed subsets contains a finite subfamily whose interiors cover \(X\). It is shown that a space is countably \(I\)-compact if and only if it is extremally disconnected and countably \(S\)-closed. Other characterizations are given in terms of covers by semiopen subsets and other types of subsets. We also show that countable \(I\)-compactness is invariant under almost open semi-continuous surjections.

MSC:

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