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Zbl 1067.54027
Berarducci, Alessandro; Dikranjan, Dikran; Pelant, Jan
Uniform quasi components, thin spaces and compact separation. (English)
[J] Topology Appl. 122, No. 1-2, 51-64 (2002). ISSN 0166-8641

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 $B$).\par 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 2000:: *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

Keywords: real-valued functions; metric spaces; quasi component; thin spaces; compact separation property

Citations: Zbl 0867.54022

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