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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extensions of continuous functions from dense subspaces


Author: Robert L. Blair
Journal: Proc. Amer. Math. Soc. 54 (1976), 355-359
MathSciNet review: 0390999
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Abstract | References | Additional Information

Abstract: Let $ X$ and $ Y$ be topological spaces, let $ S$ be a dense subspace of $ X$, and let $ f:S \to Y$ be continuous. When $ Y$ is the real line $ {\mathbf{R}}$, the Lebesgue sets of $ f$ are used to provide necessary and sufficient conditions in order that the (bounded) function $ f$ have a continuous extension over $ X$. These conditions yield the theorem of Taimanov (resp. of Engelking and of Blefko and Mrówka) which characterizes extendibility of $ f$ for $ Y$ compact (resp. realcompact). In addition, an extension theorem of Blefko and Mrówka is sharpened for the case in which $ X$ is first countable and $ Y$ is a closed subspace of $ {\mathbf{R}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0390999-0
PII: S 0002-9939(1976)0390999-0
Keywords: Continuous function, real-valued continuous function, continuous extension, dense subspace, compact space, realcompact space, zero-set, Lebesgue set, $ {C^{\ast}}$embedded, $ C$-embedded, first countable space
Article copyright: © Copyright 1976 American Mathematical Society