Weakly linearly lindelöf monotonically normal spaces are lindelöf

Abstract

A space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U. We also introduce the concept of 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 that every weakly Lindelöf monotonically normal space is Lindelöf, a result obtained earlier in [3]. We show that, under the hypothesis 2ω < ωω, if the co-diagonal ΔcX = (X × X) \ΔX is discretely Lindelöf, then X is Lindelöf and has a weaker second countable topology; here ΔX = {(x, x): x ∈ X} is the diagonal of the space X. Moreover, discrete Lindelöfness of ΔcX together with the Lindelöf Σ-property of X imply that X has a countable network.