A note on monotonically normal spaces

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 {Fnα : n < ω, α < κ} of X such that Fnα ⊂ Uα for n < ω, α < κ and PreliminariesThe B-property is introduced by P. Zenor in [10]. A space X is said to have the B-property if every increasing open cover {U α : α < κ} of X has an increasing open cover {V α : α < κ} of X such that V α ⊂ U α for each α < κ, where by {U α : α < κ} being increasing we mean U α ⊂ U β wherever α < β. It is well-known that paracompactness ⇒ B-property ⇒ countable paracompactness. But the implications are not reversible (cf. [5], [6], [8]).

The Collins–Roscoe mechanism and D-spaces

“…We mention that Y. Z. Gao, H. Z. Qu and S. T.Wang gave an interesting characterization for monotonically normal paracompact spaces in [8].…”