Finitely chainable and totally bounded metric spaces: Equivalent characterizations

A metric space (X,d) is called finitely chainable if for every ϵ>0, there are finitely many points p1,p2,...,pr in X and a positive integer m such that every point of X can be joined with some pj, 1≤j≤r by an ϵ-chain of length m. In 1958, Atsuji proved: a metric space (X,d) is finitely chainable if and only if every real-valued uniformly continuous function on (X,d) is bounded. In this paper, we study twenty-five equivalent characterizations of finitely chainable metric spaces, out of which three are entirely new. Here we would like to mention that this study essentially turns the first part of this paper into a sort of an expository research article. A totally bounded metric space is finitely chainable. In order to have a better perception of the difference between total boundedness and finite chainability, several new equivalent characterizations of totally bounded metric spaces are also studied. Moreover, two topological characterizations of metric spaces admitting compatible finitely chainable metrics are given.