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Extension of continuous functions into uniform spaces


Author: Salvador Hernández
Journal: Proc. Amer. Math. Soc. 97 (1986), 355-360
MSC: Primary 54C20; Secondary 54C30, 54D60
DOI: https://doi.org/10.1090/S0002-9939-1986-0835898-9
MathSciNet review: 835898
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Abstract: Let $ X$ be a dense subspace of a topological space $ T$, let $ Y$ be a uniformizable space, and let $ f:X \to Y$ a continuous map. In this paper we study the problem of the existence of a continuous extension of $ f$ to the space $ T$. Thus we generalize basic results of Taimanov, Engelking and Blefko-Mrówka on extension of continuous functions. As a consequence, if $ \mathcal{D}$ is a nest generated intersection ring on $ X$, we obtain a necessary and sufficient condition for the existence of a continuous extension to $ v (X,\mathcal{D})$, of a continuous function over $ X$, when the range of the map is a uniformizable space, and we apply this to realcompact spaces. Finally, we suppose each point of $ T\backslash X$ has a countable neighbourhood base, and we obtain a generalization of a theorem by Blair, herewith giving a solution to a question proposed by Blair.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0835898-9
Article copyright: © Copyright 1986 American Mathematical Society

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