What is the topological classification of spaces with non-constant cauchy sequences?

To be clear, I'm not asking about completeness of the space, just if there is some classification that can be placed on spaces where there exists a Cauchy sequence that is (eventually) non-constant. There are some trivial properties such a space has to have:

1. non-finite
2. distance

Clearly these terms are not enough to classify such a space as the natural numbers act as a counter example, so I'm curious as to what's missing. Thanks for the help!