Systems of Continuos Functions

Abstract

Topological spaces were introduced in the first place because they are the natural habitat for continuous functions.

Given two topological spaces X and Y , the number of continuous functions from X to Y can vary greatly, depending on the topologies involved: everything is possible from “all functions are continuous” to “only the constants are continuous.” In this chapter, we are interested in continuous functions from topological spaces into R or C. On a metric space, the metric itself easily provides a plentiful supply of such functions. But can anything meaningful be said in the absence of a metric?