E-3 - Modern Metrization Theorems

Abstract

Bing's Theorem states that a space X is metrizable if and only if (iff) it is collection wise normal and developable. This chapter discusses base conditions, neighborhood assignments, collection wise normal spaces, Gδ –diagonals, and point-countable bases. It emphasizes the important role of base conditions in metrization theory and highlights the variations of a σ -locally finite or a σ -discrete base. Neighborhood assignments have been a very active area of research in metrization theory, mainly because so many generalizations of metric spaces can be characterized in terms of neighborhood assignments. Bing's Theorem factors metrizability into two parts: collection wise normality and developability. There is a similar factorization in which collection wise normality is strengthened to paracompactness but developability is replaced by a weaker property. A space X is metrizable iff it is a monotonically normal p-space with a Gδ -diagonal. A point-countable collection of subsets of a set X has only countably many finite irreducible subcovers. Two classes of spaces are considered paracompact: p-spaces and M-spaces. Both p-spaces and M-spaces generalize metric spaces. Every compact space is a paracompact p-space and every countably compact space is an M-space.