PARACOMPACT SUBSETS

Abstract

This chapter presents 3 types of paracompact subsets and 2 types of countably paracompact subsets. The term paracompact subsets generally refer to β-paracompact subsets. Every α-paracompact set is σ-paracompact and every σ-paracompact set in a regular space is β-paracompact. The chapter presents a proof of the following: A β-paracompact subset of a regular normal space is σ-paracompact if it is α-collectionwise normal and generalized Fσ; in a regular normal space, closed σ-paracompact subsets are α-paracompact. To prove the latter result, it will first be proved that in a normal space a closed β-countably paracompact subset is α-countably paracompact. It will also be proved that closed subsets of the interior of β-paracompact subsets in normal spaces are α-paracompact. The α-paracompact subsets will be shown to behave in many regard as compact subsets. For instance in a T2 space, two disjoint α-paracompact subsets are strongly separated, σ-paracompact subsets have certain similarities to Lindelof subsets. In the definition of σ-paracompact subsets, σ-locally finite may be replaced by σ-discrete in regular normal spaces.