On the cardinality of almost discretely Lindelof spaces

TL;DR

This paper investigates the cardinality bounds of almost discretely Lindelöf spaces, proving they are at most continuum under certain set-theoretic assumptions and for Urysohn spaces within ZFC, thus extending previous results.

Contribution

It establishes new cardinality bounds for almost discretely Lindelöf spaces in ZFC and under set-theoretic assumptions, improving prior results for specific classes.

Findings

Cardinality at most continuum under $2^{<rak{c}}=\frak{c}$

Cardinality bounds for Urysohn spaces in ZFC

Extension of previous results by Juhász, Soukup, and Szentmiklóssy

Abstract

A space is said to be "almost discretely Lindel\"of" if every discrete subset can be covered by a Lindel\"of subspace. Juh\'asz, Tkachuk and Wilson asked whether every almost discretely Lindel\"of first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $2^{<\mathfrak{c}}=\mathfrak{c}$ (which is a consequence of Martin's Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juh\'asz, Soukup and Szentmikl\'ossy. We conclude with a few related results and questions.