Weakly linearly Lindel\"of monotonically normal spaces are Lindel\"of

TL;DR

This paper introduces weakly linearly Lindel"of spaces and proves that weakly Lindel"of monotonically normal spaces are Lindel"of, also exploring conditions under which spaces are Lindel"of based on properties of their diagonals.

Contribution

It establishes that weakly Lindel"of monotonically normal spaces are Lindel"of and explores conditions involving diagonals that imply Lindel"ofness.

Findings

Weakly Lindel"of monotonically normal spaces are Lindel"of.

Under certain set-theoretic hypotheses, spaces with discretely Lindel"of co-diagonals are Lindel"of.

Discrete Lindel"ofness of the co-diagonal and Lindel"of $ ext{Sigma}$-property imply a countable network.

Abstract

We call a space $X$ {\it weakly linearly Lindel\"of} 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 Lindel\"of} spaces as the ones in which every discrete subspace can be covered by a Lindel\"of subspace. We prove that, in addition to linearly Lindel\"of spaces, both weakly Lindel\"of spaces and almost discretely Lindel\"of spaces are weakly linearly Lindel\"of. The main result of the paper is formulated in the title. It implies, among other things, that every weakly Lindel\"of monotonically normal space is Lindel\"of; 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 $\Delta^c_X=(X\times X)\setminus \Delta_X$ of a space $X$ is discretely Lindel\"of, then $X$ is Lindel\"of 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 Lindel\"ofness of $\Delta^c_X$ together with the Lindel\"of $\Sigma$-property of $X$ imply that $X$ has a countable network.