IV - Mappings

Abstract

This chapter discusses the concept of mapping in topological spaces. One can discover and establish the relationships between two topological spaces (R, τ) and (R', τ') by examining maps from R into R'. Since (R', τ) is not to be thought of as a set but rather as a set carrying a certain topology, it behooves one to only consider maps from R into R' that are, in a certain sense, compatible with the topologies τ and τ'. The most important class of such maps is the class of continuous maps. The chapter discusses general principles for creating new topological spaces out of old ones with the aid of the theory of continuous maps. A special subclass of continuous maps allows one to define an appropriate notion of isomorphism between topological spaces. In general, an isomorphism between two sets is a 1–1 mapping between these sets preserving whatever structure one is interested in studying. In algebra, this preservation of structure is achieved by demanding that the map in question commute with the underlying algebraic structures. Likewise, an order isomorphism commutes with the order relations on the underlying sets. Topological isomorphism can also be characterized by appropriate commutativity conditions.