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

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.