Which connected metric spaces are compact?

Abstract

A metric space X X is called chainable if for each ε > 0 \varepsilon > 0 each two points in X X can be joined ε \varepsilon -chain. X X is called uniformly chainable if for ε \varepsilon there exists an integer n n such that each two points can be joined ε \varepsilon -chain of length at most n n . Theorem. A chainable metric space X X is a continuum if and only if X X is uniformly chainable and there exists δ > 0 \delta > 0 such that each closed δ \delta -ball is compact. Using Ramsey’s Theorem a sequential characterization of uniformly chainable metric spaces is obtained, paralleling the one for totally bounded spaces.