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Movability and limits of polyhedra

V. LagunaM. MoronNhu NguyenJ. Sanjurjo — 1993

Fundamenta Mathematicae

We define a metric d S , called the shape metric, on the hyperspace 2 X of all non-empty compact subsets of a metric space X. Using it we prove that a compactum X in the Hilbert cube is movable if and only if X is the limit of a sequence of polyhedra in the shape metric. This fact is applied to show that the hyperspace ( 2 2 , dS) i s s e p a r a b l e . O n t h e o t h e r h a n d , w e g i v e a n e x a m p l e s h o w i n g t h a t 2ℝ2 i s n o t s e p a r a b l e i n t h e f u n d a m e n t a l m e t r i c i n t r o d u c e d b y B o r s u k .

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