Proposition 7

Let $\mathfrak{U}_1, \mathfrak{U}_2$ be two uniformities on a set $X$, and let $\mathfrak{T}_1, \mathfrak{T}_2$ be the topologies induced by these uniformities respectively. Suppose that $\mathfrak{U}_1$ is finer than $\mathfrak{U}_2$, and that there is a fundamental system of entourages for $\mathfrak{U}_1$ which are closed in $X \times X$ in the topology $\mathfrak{T}_2 \times \mathfrak{T}_2$. Then a filter $\mathfrak{F}$ on $X$ converges in the topology $\mathfrak{T}_1$ if and only if it is a Cauchy filter in the uniformity $\mathfrak{U}_1$ and converges in the topology $\mathfrak{T}_2$.

The conditions are clearly necessary, because $\mathfrak{T}_2$ is coarser than $\mathfrak{T}_1$. Conversely, suppose that the conditions are satisfied, and let $x$ be a limit point of $\mathfrak{F}$ with respect to $\mathfrak{T}_2$; we shall show that $x$ is a limit of $\mathfrak{F}$ with respect to $\mathfrak{T}_1$. Let $V$ be a symmetric entourage of $\mathfrak{U}_1$ which is closed in the topology $\mathfrak{T}_2 \times \mathfrak{T}_2$. By hypothesis, $\mathfrak{F}$ contains a set $M$ which is V-small; hence if $x' \in M$ we have $M \subset V(x')$. But $V(x')$ is closed in the topology $\mathfrak{T}_2$; hence $x$, which lies in the closure of $M$ with respect to $\mathfrak{T}_2$, must belong to $V(x')$. It follows that $M \subset \hat{V}(x)$, and the proposition is proved.