Uniform quasi components, thin spaces and compact separation


Alessandro Berarducci, Dikran Dikranjan, Jan Pelant

14 pages

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 is thin. The UA-spaces form a class properly including the Atsuji spaces.

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