A generalization of the strict topology

Giles, Robin

doi:10.1090/S0002-9947-1971-0282206-4

A generalization of the strict topology
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Trans. Amer. Math. Soc. 161 (1971), 467-474 Request permission

Abstract:

The strict topology $\beta$ on the space $C(X)$ 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) $C(X)$ is $\beta$-complete, (b) the dual of $C(X)$ under the strict topology is the space of all finite signed regular Borel measures on X, and (c) a Stone-Weierstrass theorem holds for $\beta$-closed subalgebras of $C(X)$. 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.

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