*(English)*Zbl 1067.54027

Summary: We prove that every complete metric space $X$ 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 $A$ and $B$ of $X$ there is a compact set $K$ disjoint from $A$ and $B$ such that every neighbourhood of $K$ disjoint from $A$ and $B$ separates $A$ and $B$).

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 $UA$-space [see the first two authors, Rend. Inst. Mat. Univ. Trieste 25, 23–55 (1993; Zbl 0867.54022)] is thin. The $UA$-spaces form a class properly including the Atsuji spaces.

##### MSC:

54F55 | Unicoherence, multicoherence |

54C30 | Real-valued functions on topological spaces |

41A30 | Approximation by other special function classes |

54E35 | Metric spaces, metrizability |

54D15 | Higher separation axioms |