Some covering properties for -spaces.
Weak extent in normal spaces
If is a space, then the of is the cardinal If is an open cover of , then there exists such that and . In this note, we show that if is a normal space such that and , then does not have a closed discrete subset of cardinality . We show that this result cannot be strengthened in ZFC to get that the extent of is smaller than , even if the condition that is replaced by the stronger condition that is separable.
On monotone Lindelöfness of countable spaces
A space is monotonically Lindelöf (mL) if one can assign to every open cover a countable open refinement so that refines whenever refines . We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.
Spaces with --linked topologies as special subspaces of separable spaces
We characterize spaces with --linked bases as specially embedded subspaces of separable spaces, and derive some corollaries, such as the -productivity of the property of having a -linked base.
Sequential + separable vs sequentially separable and another variation on selective separability
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.
Centered-Lindelöfness versus star-Lindelöfness
We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.
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