Weakly linearly Lindelöf monotonically normal spaces are Lindelöf

Abstract

We call a space $X$ {\it weakly linearly Lindelöf} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $\kappa$, there exists a point $x\in X$ such that every neighborhood of $x$ meets $\kappa$-many elements of $\mathcal{U}$. We also introduce the concept of {\it almost discretely Lindelöf} spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the paper is formulated in the title. It implies, among other things, that every weakly Lindelöf monotonically normal space is Lindelöf; this result seems to be new even for linearly ordered topological spaces. We show that, under the hypothesis $2^{\omega }<{\omega }_{\omega }$, if the co-diagonal ${\Delta }_X^c=(X\times X)\setminus {\Delta }_X$ of a space $X$ is discretely Lindelöf, then $X$ is Lindelöf and has a weaker second countable topology; here ${\Delta }_X=\{(x,x):x\in X\}$ is the diagonal of the space $X$. Moreover, the discrete Lindelöfness of ${\Delta }_X^c$ together with the Lindelöf $\Sigma$-property of $X$ imply that $X$ has a countable network.

13 pages