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On monotone Lindelöfness of countable spaces

Ronnie LevyMikhail Matveev — 2008

Commentationes Mathematicae Universitatis Carolinae

A space is monotonically Lindelöf (mL) if one can assign to every open cover 𝒰 a countable open refinement r ( 𝒰 ) so that r ( 𝒰 ) refines r ( 𝒱 ) whenever 𝒰 refines 𝒱 . We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.

Weak extent in normal spaces

Ronnie LevyMikhail Matveev — 2005

Commentationes Mathematicae Universitatis Carolinae

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

Sequential + separable vs sequentially separable and another variation on selective separability

Angelo BellaMaddalena BonanzingaMikhail Matveev — 2013

Open Mathematics

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

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