Search Results
Abstract
We show that a monotonically normal space X is paracompact if and only if for every increasing open cover {U α : α < κ} of X, there is a closed cover {F nα : n < ω, α < κ} of X such that F nα ⊂ U α for n < ω, α < κ and F nα ⊂ F nβ if α ≦ β.
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 < ω ω, if the co-diagonal Δ c X = (X × X) \Δ X is discretely Lindelöf, then X is Lindelöf and has a weaker second countable topology; here Δ X = {(x, x): x ∈ X} is the diagonal of the space X. Moreover, discrete Lindelöfness of Δ c X together with the Lindelöf Σ-property of X imply that X has a countable network.
Abstract
Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality.
spaces Pacific J. Math. 81 371 – 377 . [8] Gao , Y. Z. , Qu , H. Z. , Wang , S. T. 2007 A note on monotonically normal spaces Acta Math. Hungar. 117 175 – 178 10.1007/s10474
