A note on rings of continous functions

J S Yang

doi:10.1155/s0161171278000113

Abstract

For a topological spaceX, and a topological ringA, letC(X,A)be the ring of all continuous functions fromXintoAunder the pointwise multiplication. We show that the theorem there is a completely regular spaceYassociated with a given topological spaceXsuch thatC(Y,R)is isomorphic toC(X,R) may be extended to a fairly large class of topologlcal rings, and that, in the study of algebraic structure of the ringC(X,A), it is sufficient to studyC(X,R)ifAis path connected.

Highlights

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
C(X,A) considered there are the topology of pointwise convergence, the compact-open topology, and the topology of uniform convergence
In the study of rings of real-valued continuous functions on a topological space, it is usually assumed that X is completely regular

Summary

Introduction

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. In the study of rings of real-valued continuous functions on a topological space, it is usually assumed that X is completely regular. For every topological space X, there exists a completely regular space Y such that C(Y) is (algebraically) isomorphic to C(X). DEFINITION A pair (X,A) of a topological space X and a topological ring

Results

Conclusion