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

Abstract

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. MSC:54D65, 54A25, 54D55, 54A20 Mikhail "Misha" Matveev passed away unexpectedly on

Sequential separability vs selective sequential separability

“…Our second example shows that this is not possible, even within the class of countable spaces. This example also answers Question 4.10 of [3]. That question for a large sequential separable space was implicitely answered in [3]…”

“…In [3], Question 4.3, it is asked to find a sequentially separable selectively separable space which is not selectively sequentially separable. But surprisingly, even the very basic question on the existence of a sequentially separable space which is not selectively sequentially separable is uncovered there.…”

“…However, notice that the spaces with a countable π-base are much more than selectively separable. Example 2.1 in [3] provides a compact T 2 sequential space which is not selectively sequentially separable. The trivial reason is because that space is not sequentially separable.…”

Strongly sequentially separable function spaces, via selection principles

“…A separable space is strongly sequentially separable (SSS) [9] if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. Every separable Fréchet-Urysohn space is strongly sequentially separable, but not conversely [1,Example 2.4].…”

Diagonalizations of dense families

“…A [27], S fin (D, D) [3], M-separable [7,8,22] selectively separable (SS) [10,3,4,16,5,13,17,23,9,6]…”