A generalization of the strict topology

Abstract

The strict topology β \beta on the space C ( X ) C(X) of bounded real-valued continuous functions on a topological space X was defined, for locally compact X, by Buck (Michigan Math. J. 5 (1958), 95-104). Among other things he showed that (a) C ( X ) C(X) is β \beta -complete, (b) the dual of C ( X ) C(X) under the strict topology is the space of all finite signed regular Borel measures on X, and (c) a Stone-Weierstrass theorem holds for β \beta -closed subalgebras of C ( X ) C(X) . In this paper the definition of the strict topology is generalized to cover the case of an arbitrary topological space and these results are established under the following conditions on X: for (a) X is a k-space; for (b) X is completely regular; for (c) X is unrestricted.