Equivalent definitions of complete uniform space

Question

A uniform space $(X,\mathcal{U})$ is called complete if every Cauchy filter converges.

In this page, Brian M.Scott says that,

In a uniform space every Cauchy filter converges iff every Cauchy net converges;

that is, uniform space $(X,\mathcal{U})$ is complete if every Cauchy net converges. I can show that if a uniform space is complete then every Cauchy net converges. My problem is proving the converse. Thanks for any ideas.

Answer 1

Suppose $(X,\mathcal D)$ is a (diagonal) uniform space.

Every net on the set $X$ of the form $n:(D,\le)\to X$, where $(D,\le)$ is a directed set, has a corresponding filter $\mathcal F_n$ generated by the filter-base (=centered-family): $$\lbrace \lbrace n(x)\mid x\ge d \rbrace \mid d\in D\rbrace$$

It can be shown that

1. Any filter on $X$ is of the form $\mathcal F_n$ for some net $n$ on $X$.
2. A net $n$ on X is Cauchy in $(X,\mathcal D)$ iff the filter $\mathcal F_n$ is Cauchy in $(X,\mathcal D)$.
3. A net $n$ on X is converges to a point $a$ in $(X,\mathcal D)$ iff the filter $\mathcal F_n$ is converges to $a$ in $(X,\mathcal D)$.

Now it's clear that these propositions are equivalent:

1. Any Cauchy filter in $(X,\mathcal D)$ is convergent.
2. Any Cauchy filter in $(X,\mathcal D)$ of the form $\mathcal F_n$, where $n$ is a net on $X$, is convergent.
3. Any filter in $(X,\mathcal D)$ of the form $\mathcal F_n$, where $n$ is a Cauchy net on $X$, is convergent.
4. Any Cauchy net in $(X,\mathcal D)$ is convergent.