Completely Regular Spaces

Abstract

Up to this point in the text, we have not assumed any separation axioms for the topological space on which our ring of continuous functions is defined. Indeed, separation axioms were irrelevant to most of the subjects discussed. We have now reached the stage where separation properties of the space do enter in an essential way, so that we are forced to make a decision about what class or classes of spaces to consider. We have no desire to become involved in finding the weakest axiom under which each theorem can be proved, but prefer, if possible, to stick to a single class of topological spaces that is wide enough to include all of the interesting spaces, and, at the same time, restrictive enough to admit a significant theory of rings of continuous functions.