A note on rings of continous functions

Abstract

For a topological space X, and a topological ring A, let C(X,A) be the ring of all continuous functions from X into A under the pointwise multiplication. We show that the theorem “there is a completely regular space Y associated with a given topological space X such that C(Y,R) is isomorphic to C(X,R)” may be extended to a fairly large class of topologlcal rings, and that, in the study of algebraic structure of the ring C(X,A), it is sufficient to study C(X,R) if A is path connected.

Keywords

• continuous functions
• completely regular space
• topological ring
• S-pair
• compact-open topology.