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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 in general topology 41A30 Approximation by other special function classes 54E35 Metric spaces, metrizability 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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