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 [Rend. Instit. Mat. Univ. Trieste 25 (1993) 23–56]) is thin. The UA-spaces form a class properly including the Atsuji spaces.
Uniform quasi components, thin spaces and compact separation
BERARDUCCI, ALESSANDRO;
2002-01-01
Abstract
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 [Rend. Instit. Mat. Univ. Trieste 25 (1993) 23–56]) is thin. The UA-spaces form a class properly including the Atsuji spaces.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
