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Ronnie Levy, Mikhail Matveev 9A= ANJAJ E H= IF=?AI

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Ronnie Levy, Mikhail Matveev Weak extent in normal spaces

Comment.Math.Univ.Carolinae 46,3 (2005) 497-501.

Abstract: If X is a space, then the weak extent we(X) of X is the cardinal min{α: IfU is an open cover ofX, then there existsA⊆X such that|A|=αand St(A,U) =X}. In this note, we show that ifXis a normal space such that|X|=c andwe(X) =ω, thenX does not have a closed discrete subset of cardinalityc. We show that this result cannot be strengthened in ZFC to get that the extent ofX is smaller thanc, even if the condition thatwe(X) =ω is replaced by the stronger condition thatX is separable.

Keywords: extent, weak extent, separable, star-Lindel¨of, normal AMS Subject Classification: Primary 54A25, 54D40

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