Quotients of Metric Spaces

Abstract

The question of when a given property of a topological space is preserved under mappings is one of the most familiar problems of general topology. Among the properties of greatest interest for general spaces is, without doubt, that of metrizability; metrizability always implies, in particular, a number of important special topological properties of the space in question (normality, regularity, paracompactness, etc.). To determine in general the conditions for preservation of metrizability under mappings appears to be a difficult problem. It may well be that in the class of arbitrary continuous mappings the problem has no meaningful solution. The purpose of this paper is to obtain conditions for the preservation of metrizability under mappings appears to be a difficult problem. It may well be that in the class of arbitrary continuous mappings the problem has no meaningful solution. The purpose of this paper is to obtain conditions for the preservation of metrizability by quotient mappings and to study the properties of quotient spaces of metric spaces. We will use "iff" as an abbreviation for "if and only if". If f is a function from X onto Y, we will write f: X --->> Y.