Monotonically normal space

Abstract: We obtain various characterizations of monoton- ically normal spaces which not only answer various questions of Zenor but also allow an elementary proof of a result of Heath and Lutzer. We also prove that elastic spaces are monotonically normal. Recently P. Zenor introduced the class of monotonically normal spaces (his results will appear in (3)). Soon afterward, R. Heath and D. Lutzer proved that each linearly ordered topological space is monotonically normal (see (3)). Since Zenor did not know if monotonically normal spaces were hereditarily monotonically normal and the result of Heath and Lutzer had a very long proof, my interest in the study of these spaces was aroused. The characterizations of monotonically normal spaces which follow not only answer Zenor's questions but also permit us to prove, quite easily, that linearly ordered topological spaces and elastic spaces are monotonically normal. (Our proof of the first result appears in (2). For the second result, see Theorem 2.3.) 1. Characterizations of monotonically normal spaces. For the sake of completeness, we will first define this class of spaces. Definition 1.1. For any space X, let 3)X={(A, B)\A and B are disjoint closed subsets of A"}. The Trspace X is said to be monotonically normal provided that to each (A, B) e @)x one can assign an open subset G(A, B) of X such that (a) A<=G(A,B)<=G(A,B)-^X-B, (b) G(A, B)^G(A', B'), whenever A <=A' and B'<=B. The function G is called a monotone normality operator.

Abstract: A space X is weakly linearly Lindelof 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 Lindelof spaces as the ones in which every discrete subspace can be covered by a Lindelof subspace. We prove that, in addition to linearly Lindelof spaces, both weakly Lindelof spaces and almost discretely Lindelof spaces are weakly linearly Lindelof. The main result of the paper is formulated in the title. It implies that every weakly Lindelof monotonically normal space is Lindelof, a result obtained earlier in [3]. We show that, under the hypothesis 2ω < ωω, if the co-diagonal ΔcX = (X × X) \ΔX is discretely Lindelof, then X is Lindelof and has a weaker second countable topology; here ΔX = {(x, x): x ∈ X} is the diagonal of the space X. Moreover, discrete Lindelofness of ΔcX together with the Lindelof Σ-property of X imply that X has a count...