On [formula omitted]-discrete, T-finite and tree-type topologies

Abstract

A regular space is T-finite if and only if it is hereditarily strongly collectionwise Hausdorff and σ-pseudo-closed discrete. Every finer regular topology on such a space is hereditarily ultraparacompact. σ-pseudo-closed discreteness is strictly between σ-closed discreteness and σ-discreteness. It yields ultraparacompactness for regular, strongly collectionwise Hausdorff spaces. Every T-finite, regular topology is finer than a (more or less) canonical topology defined on a tree of height ≤ω. These tree-type topologies (for arbitrary height) are always ultraparacompact and monotonically normal. A space is non-Archimedean and left separated if and only if it is a lob and of tree-type.