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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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