Summary: We prove that every complete metric space that is thin (i.e., every closed subspace has connected uniform quasi components) has the compact separation property (for any two disjoint closed connected subspaces and of there is a compact set disjoint from and such that every neighbourhood of disjoint from and separates and ).
The real line and all compact spaces are obviously thin. We show that a space is thin if and only if it does not contain a certain forbidden configuration. Finally, we prove that every metric -space [see the first two authors, Rend. Inst. Mat. Univ. Trieste 25, 23–55 (1993; Zbl 0867.54022)] is thin. The -spaces form a class properly including the Atsuji spaces.