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A characterization of $ Z$-separating algebras


Author: Shankar Hegde
Journal: Proc. Amer. Math. Soc. 73 (1979), 40-44
MSC: Primary 46J10; Secondary 54C50
DOI: https://doi.org/10.1090/S0002-9939-1979-0512055-3
MathSciNet review: 512055
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Abstract: Let A be a uniformly closed point separating algebra of bounded real valued functions on a set X, containing the constant functions. A is called z-separating if whenever $ {Z_1},{Z_2}$ are disjoint zero sets of members of A there is some $ f \in A$ with $ f({Z_1}) = 0$ and $ f({Z_2}) = 1$. We prove that A is z-separating if and only if A consists of precisely those bounded real valued functions f on X for which $ {f^{ - 1}}(C)$ is a zero set of some member of A for every closed set C of real line.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0512055-3
Keywords: z-separating algebra, Wallman compact space, $ \mathcal{L}$-continuity and $ \mathcal{L}$-uniform continuity
Article copyright: © Copyright 1979 American Mathematical Society