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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A generalization of the strict topology

Author: Robin Giles
Journal: Trans. Amer. Math. Soc. 161 (1971), 467-474
MSC: Primary 46.25
MathSciNet review: 0282206
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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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Keywords: Strict topology, Stone-Weierstrass theorem, completely regular space, k-space, regular Borel measure
Article copyright: © Copyright 1971 American Mathematical Society