When is a uniform space complete

From Wikipedia:

a uniform space is called complete if every Cauchy filter converges.

1. I was wondering if the following three are equivalent in a uniform space:

   • every Cauchy filter converges,
   • every Cauchy net converges, and
   • every Cauchy sequence converges?

   Or, which one implies which but doesn't imply which? For example, are the first two equivalent, while the third is implied by but does not implie any of the first two?

2. How about in a metric space?

Thanks and regards!

In a metric space all three properties are equivalent.

In a uniform space every Cauchy filter converges iff every Cauchy net converges; the usual equivalence between filters and nets in arbitrary topological spaces preserves the property of being Cauchy in uniform spaces. This is strictly stronger than merely requiring Cauchy sequences to converge.

Example: Let $X$ be $\omega_1$ with the order topology. $X$ is Tikhonov, so it has a compatible uniformity $\mathscr{U}$. $X$ is countably compact, so $\langle X,\mathscr{U}\rangle$ is totally bounded, but $X$ is not compact, so $\langle X,\mathscr{U}\rangle$ is not complete: a uniform space is compact iff it is complete and totally bounded. Thus, $\langle X,\mathscr{U}\rangle$ must have Cauchy filters/nets that do not converge. However, every sequence in $X$ is contained in a compact subspace of $X$, so every Cauchy sequence does converge.