First countable and almost discretely Lindelöf $T_3$ spaces have cardinality at most continuum

Abstract

A topological space $X$ is called almost discretely Lindelöf if every discrete set $D\subset X$ is included in a Lindelöf 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 Lindelöf $T_3$ 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. Juhász, V. Tkachuk, and R. Wilson, Weakly linearly Lindelöf monotonically normal spaces are Lindelöf", 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 Lindelöf spaces".

arXiv comment7 pages