Separable extensions of first countable spaces
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.
Small subsets of first countable spaces
Spaces with a locally countable -network.
Spaces with countable -networks
In this paper, we prove that a space is a sequentially-quotient -image of a metric space if and only if has a point-star -network consisting of -covers. By this result, we prove that a space is a sequentially-quotient -image of a separable metric space if and only if has a countable -network, if and only if is a sequentially-quotient compact image of a separable metric space; this answers a question raised by Shou Lin affirmatively. We also obtain some results on spaces with 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.
Stationary sets, trees and continuums.