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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:
54F55Unicoherence, multicoherence
54C30Real-valued functions on topological spaces
41A30Approximation by other special function classes
54E35Metric spaces, metrizability
54D15Higher separation axioms