CHAPTER 9 - Function and Quotient Spaces

Abstract

This chapter describes the function and quotient spaces. It explains the concept of compact – open topology or the topology of compact convergence. The topology of uniform convergence on compact is the same as the compact-open topology for the set of all bounded, continuous functions from the topological space to the metric space. As compact sets have many of the nice properties of finite sets, therefore, the topology induced by the family of all compact subsets is called the compact-open topology or the topology of compact convergence. Every pseudometric space is a completely normal first axiom space.