An introduction to shape theory for the nonspecialist

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 $(2^{ℝ} 2$ , 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 $i s n o t s e p a r a b l e i n t h e f u n d a m e n t a l m e t r i c i n t r o d u c e d b y B o r s u k .$