Which connected metric spaces are compact?

Abstract:

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