On σ-discrete, T-finite and tree-type topologies

✍ Scribed by Ulrich Heckmanns; Stephen Watson

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 163 KB
• Volume: 101
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00097-2

✦ Synopsis

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 σdiscreteness. 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.