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On removing one point from a compact space. (English) Zbl 1069.54017

A space \(X\) is called {star-Lindelöf} if for every open cover \( \mathcal{U}\) of \(X\) there exists a countable set \(F\subset X\) such that \( st(F,\mathcal{U})=X\). If the set \(F\) can be chosen to be closed and discrete as well as countable, then \(X\) is called {discretely} { star-Lindelöf.} The property of a space being star-Lindelöf is a rather strange property in that it generalizes {both} countably compact and Lindelöf, rather unusual for something that looks like a covering property.
In the paper under review, the author studies the question of the preservation of the star-Lindelöf property when a point is removed from a compact space, or from a product of compact spaces. The author proves that if \(B\) is a compact space and \(0\in B\) such that \(B\backslash \{0\}\) is Lindelöf, then for any cardinal \(\kappa \) the space \(B^{\kappa }\backslash \{\overrightarrow{0}\}\) is star-Lindelöf. He also proves that if \(B\) is a \(T_{1}\)-space and \(0\in B\) such that \(B\backslash \{0\}\) is compact, then for any cardinal \(\kappa \) the space \(B^{\kappa }\backslash \{ \overrightarrow{0}\}\) is discretely star-Lindelöf.
Several other interesting results are reported; for example, the space \(\{0,1\}^{\kappa }\backslash \{pt\}\) is pseudocompact whenever \(\kappa \) is an uncountable cardinal.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
54B10 Product spaces in general topology

Keywords:

star-Lindelöf
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