Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Correction: Proc. Amer. Math. Soc. 106 (1989), 857-858.
MathSciNet review: 0390999
Full-text PDF

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}}$.

References [Enhancements On Off] (What's this?)

Additional Information

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

American Mathematical Society