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\begin{proposition}[{{\cite[Proposition 2.3.11]{Bourbaki}}}]
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\label{proposition:uniformextension}
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Let $X$ be a topological space, $Y$ be a complete Hausdorff uniform space, $A \subset X$ be a dense subset, and $f \in C(A; Y)$ be a continuous function, then the following are equivalent:
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Let $X$ be a uniform space, $Y$ be a complete Hausdorff uniform space, $A \subset X$ be a dense subset, and $f \in C(A; Y)$ be a continuous function, then the following are equivalent:
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\begin{enumerate}
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\item There exists a continuous function $F \in C(X; Y)$ such that $F|_A = f$.
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\item $f$ is Cauchy continuous.
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