Proposition 11

*Let $A$ be a dense subset of a topological space $X$, and let $f$ be a mapping of $A$ into a complete Hausdorff uniform space $X'$. Then $f$ can be extended by continuity to $X$ if and only if, for each $x \in X$, the image under $f$ of the trace on $A$ of the neighbourhood filter of $x$ in $X$ is a Cauchy filter base in $X'$.

This follows from the theorem of extension by continuity (loc. cit.) because $X'$ is regular (§ 1, no. 2, Proposition 3) and because on $X'$ convergent filters are the same as Cauchy filters.

When $X$ is also a uniform space, there is the following theorem: