On monotonically normal and transitively (linearly) dually discrete spaces

Abstract

In this article, we show that if X is a monotonically normal space, then for any neighborhood assignment ϕ for X there exists a discrete subspace D of X such that X=(⋃{ϕ(d):d∈D})∪D‾ and D‾∖(⋃{ϕ(d):d∈D}) is left-separated. A space X is called weakly dually discrete if for any neighborhood assignment ϕ for X there exists a discrete subspace D of X such that X=(⋃{ϕ(d):d∈D})∪D‾ and D‾∖(⋃{ϕ(d):d∈D}) is a closed discrete subspace of X. We discuss some basic properties of weakly dually discrete spaces.In the last part of this article, we introduce the notions of linear dual discreteness and transitive dual discreteness. Some of their properties are discussed. We finally show that if a space X is discretely complete and X=⋃{Xn:n∈N} such that Xn is monotonically normal for each n∈N, then X is compact. If X is a monotonically normal space and Y is a compact T1-space, then X×Y is dually 2-scattered. We also discuss some properties of spaces which are dually scattered with finite rank.