Displaying similar documents to “Convergence in Hausdorff metric preserves geometric shape.”

Movability and limits of polyhedra

V. Laguna, M. Moron, Nhu Nguyen, J. Sanjurjo (1993)

Fundamenta Mathematicae

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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 .

Shape index in metric spaces

Francisco R. Ruiz del Portal, José M. Salazar (2003)

Fundamenta Mathematicae

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We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map...

Strong shape of the Stone-Čech compactification

Sibe Mardešić (1992)

Commentationes Mathematicae Universitatis Carolinae

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J. Keesling has shown that for connected spaces X the natural inclusion e : X → β X of X in its Stone-Čech compactification is a shape equivalence if and only if X is pseudocompact. This paper establishes the analogous result for strong shape. Moreover, pseudocompact spaces are characterized as spaces which admit compact resolutions, which improves a result of I. Lončar.