From Wikipedia:
a uniform space is called complete if every Cauchy filter converges.
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?
- How about in a metric space?
Thanks and regards!