A Note on Rings of Continuous Functions
نویسنده
چکیده
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. For a topological space X and a topological ring A, let C(X,A) denote the ring of all continuous functions from X into A under the pointwlse multiplication. If A is the ring of real numbers R with the usual topology, C(X,R) will simply be denoted by C(X). In [2], the structure of the ring C(X,A), where X is totally disconnected, is studied. The topologies for
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