Proposition 8

Let X be a set, $\mathfrak{T}$ a filter on X, and let Y be a uniform space. Then the set H of mappings $u : X \to Y$ such that $u(\mathfrak{T})$ is a Cauchy filter base on Y is closed in $F_u(X; Y)$.

Let $u_0 : X \to Y$ lie in the closure of H in $F_u(X; Y)$. For each symmetric entourage V of Y, there is a mapping $u \in H$ such that $(u_0(x), u(x)) \in V$ for all $x \in X$; on the other hand, by hypothesis there is a set $M \in \mathfrak{T}$ such that $(u(x), u(x')) \in V$ whenever $x$ and $x'$ are in M. Since $(u_0(x), u(x)) \in V$ and $(u_0(x'), u(x')) \in V$, it follows that $(u_0(x), u_0(x')) \in V$ whenever $x$ and $x'$ are in M, and therefore $u_0(\mathfrak{T})$ is a Cauchy filter base on Y.