B-4 - Quotient Spaces and Decompositions

Abstract

This chapter considers a set Y to illustrate the concept of quotient spaces and decompositions. For a topological space X and a surjection ƒ: X → Y, if T (ƒ) = {U ⊂ Y : ƒ–1(U) is open in X}, then T (ƒ) is called the quotient topology (or identification topology) for Y determined by ƒ. T (ƒ) is the largest topology on Y such that the map ƒ is continuous. If X and Y are topological spaces and if ƒ: X→Y is a surjection, then ƒ is called a quotient map (or identification map) if the topology in Y is exactly T (ƒ); that is, U is open in Y if and only if (iff) ƒ–11(U) is open in X. The space Y is called the quotient space (or quotient image) of X by ƒ. Also, ƒ is called an open map (respectively closed map) if the image of each open (respectively closed) subset of X is open (respectively closed) in Y. Every open (or closed) continuous map is a quotient map. Every one-to-one quotient map is a homeomorphism.