D-13 - Generalizations of Paracompactness

Abstract

Compactness and metrizability are the heart and soul of general topology. These two concepts are the most important from the point of view of applications. Metric notions are used almost everywhere in mathematical analysis, while compactness is used in many parts of analysis and also in mathematical logic. Paracompactness may be defined as a natural generalization of compactness. Full normality and paracompactness have been shown to be equivalent properties. A space X is countably paracompact provided that each countable open cover has a locally finite open refinement. The archetypal covering property, compactness, is preserved in Cartesian products by the famous Tychonoff Theorem, but most other covering properties do not behave well with products. Metacompactness is characterized by a condition whose relation to the defining condition is similar to the relation that full normality has with paracompactness. A space is subparacompact if every open cover of the space has a σ -discrete closed refinement. A space is meta-Lindelöf if each open cover has a point countable open refinement. The relationship between normal meta-Lindelöf spaces and paracompact spaces is even more problematic than that between normal metacompact spaces and paracompact spaces because the Michael–Nagami Theorem does not extend to meta-Lindelöf spaces.