Chain development of metric compacts

Chain distance between points in a metric space is defined as the infimum of ε such that there is an ε-chain connecting these points. We call a mapping of a metric compact into the real line a chain development if it preserves chain distances. We give a criterion of existence of the chain development for metric compacts. We prove the diameter of any chain development of a given compact to be the same iff the compact is countable.