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.
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.
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.
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.
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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