Extension of continuous functions into uniform spaces

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.