First countable and almost discretely Lindel\"of $T_3$ spaces have cardinality at most continuum

Abstract

A topological space $X$ is called almost discretely Lindel\"of if every discrete set $D \subset X$ is included in a Lindel\"of subspace of $X$. We say that the space $X$ is {\em $\mu$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le \mu$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindel\"of $T_3$ space $X$ we have $|X| \le 2^{\chi(X)}$. (2) If $X$ is a $\mu$-sequential $T_2$ space of pseudocharacter $\psi(X) \le 2^\mu$ and for every free set $D \subset X$ we have $L(\overline{D}) \le \mu$, then $|X| \le 2^\mu$. The case $\chi(X) = \omega$ of (1) provides a solution to Problem 4.5 from "I. Juh\'asz, V. Tkachuk, and R. Wilson, Weakly linearly Lindel\"of monotonically normal spaces are Lindel\"of", while the case $\mu = \omega$ of (2) is a partial improvement on the main result of "A.V. Archangel'skii and R.Z. Buzyakova, On some properties of linearly Lindel\"of spaces".