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Uniform quasi components, thin spaces and compact separation. (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