Rings of continuous functions on open convex subsets of 𝑅ⁿ

Abstract

0. Introduction. If h is a homeomorphism from a topological space X onto another space Y, then one can easily show that the correspondence f-f o h-1 is an isomorphism from C(X), the ring of all real continuous functions on X, onto C( Y), the ring of all real continuous functions on Y. The converse problem-given an isomorphism 4 from C(X) onto C(Y) to show that there exists a homeomorphism h from X onto Y such that qf =f o h-l is more difficult and is one of the problems which has motivated much of the research on rings of continuous functions. The isomorphism 4 is usually constructed by using fixed, maximal ideals. An ideal M in C(X) is fixed if there exists a point xo of X such that f(xo) = 0 for all f in M. If X is completely regular, then the set M(x) of all functions in C(X) which are zero at x is a fixed, maximal ideal and all fixed, maximal ideals are of this form. If one can show that the property of a maximal ideal being fixed is invariant under ring isomorphisms and if X and Y are completely regular, then one can show that the correspondence