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

TL;DR

This paper proves new bounds on the cardinality of certain topological spaces with specific Lindel"of and sequential properties, using elementary submodels, advancing understanding of space size constraints.

Contribution

It establishes cardinality bounds for almost discretely Lindel"of $T_3$ spaces and $ ext{ extmu}$-sequential $T_2$ spaces under certain conditions, solving open problems.

Findings

For almost discretely Lindel"of $T_3$ spaces, $|X| \,\le\, 2^{\chi(X)}$.

In $ ext{ extmu}$-sequential $T_2$ spaces, $|X| \,\le\, 2^{\mu}$ under specified conditions.

Provides partial solutions to existing open problems in topological space cardinality.

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".