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Weak extent in normal spaces. (English) Zbl 1121.54012

Summary: If \(X\) is a space, then the weak extent \(\text{we}(X)\) of \(X\) is the cardinal \(\min \{\alpha \:\) If \(\mathcal U\) is an open cover of \(X\), then there exists \(A\subseteq X\) such that \(| A| = \alpha \) and \(\text{St}(A,\mathcal U)=X\}\). In this note, we show that if \(X\) is a normal space such that \(| X| = \mathfrak c\) and \(\text{we}(X) = \omega \), then \(X\) does not have a closed discrete subset of cardinality \(\mathfrak c\). We show that this result cannot be strengthened in ZFC to get that the extent of \(X\) is smaller than \(\mathfrak c\), even if the condition that \(\text{we}(X) = \omega \) is replaced by the stronger condition that \(X\) is separable.

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D40 Remainders in general topology
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