Proposition 3

Let $f : X \to X'$ be a uniformly continuous mapping. Then the image under $f$ of a Cauchy filter base on $X$ is a Cauchy filter base on $X'$.

Let $g = f \times f$. If $V'$ is an entourage of $X'$, then $\overline{g^{-1}(V')}$ is an entourage of $X$, and the image under $f$ of a $\overline{g^{-1}(V')}$-small set is $V'$-small; hence the result.

It follows in particular that if the uniformity of a uniform space $X$ is replaced by a coarser uniformity, then every Cauchy filter with respect to the original uniformity remains a Cauchy filter with respect to the new uniformity.

This fact can be easily remembered in the following form: the finer the uniformity, the fewer Cauchy filters there are.