A characterization of 𝑍-separating algebras

Hegde, Shankar

doi:10.1090/S0002-9939-1979-0512055-3

A characterization of $Z$-separating algebras
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Proc. Amer. Math. Soc. 73 (1979), 40-44 Request permission

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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