Conclusions on Totally Bounded metric spaces and Cauchy sequences
See also https://math.stackexchange.com/questions/9964/if-no-cauchy-subsequence-exists-must-a-uniformly-separated-subsequence-exist and https://math.stackexchange.com/questions/7210/which-metric-spaces-are-totally-bounded Let's list the conclusions from my recent posts. We'll use the same definitions and notation as before. These results are presumably all well known, though I haven't personally seen explicit statements of all of them before. I had some fun working out the details for myself! Notes on terminology : I have seen both "uniformly separated" and "separated" used for sequences. But there is a completely different notion of separated sets for pairs of sets. So possibly my use of the term "separated set" is confusing. I have seen the term "uniformly discrete" used, but I don't think that I have yet found the most definitive standard terminology. Theorem 1 Let \((X,d)\) be a metric space. Then the following statements...