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Approximation of continuous functions on pseudocompact spaces


Author: C. E. Aull
Journal: Proc. Amer. Math. Soc. 81 (1981), 490-494
MSC: Primary 54C40; Secondary 54D30
DOI: https://doi.org/10.1090/S0002-9939-1981-0597669-6
MathSciNet review: 597669
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Abstract: If $ {\mathcal{S}^ * }$ is the family of subrings of $ {C^ * }(X)$ such that if $ S \in {\mathcal{S}^ * }$, $ S$ contains the constant functions and is closed under uniform convergence, then the following are equivalent for a space $ (X,\mathcal{J})$. (a) $ (X,\mathcal{J})$ is pseudocompact. (b) If $ S \in {\mathcal{S}^ * }$ functionally separates points and zero sets, $ S$ generates $ (X,\mathcal{J})$. (c) If $ S \in {\mathcal{S}^ * }$ functionally separates zero sets, $ S = {C^ * }(X)$. (d) If $ S \in {\mathcal{S}^ * }$ generates the zero sets on $ (X,\mathcal{J}),S = {C^ * }(X)$. (e) If $ f \in S \in {\mathcal{S}^ * }$ and $ Z(f) = \phi $ then $ 1/f \in S$ (even when it is required that $ S$ generate the topology). (f) If $ f \in S \in \mathcal{S}$ then $ \left\vert f \right\vert \in S$ (even when it is required that $ S$ generate the topology).


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DOI: https://doi.org/10.1090/S0002-9939-1981-0597669-6
Article copyright: © Copyright 1981 American Mathematical Society

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