The completion of the hyperspace of finite subsets, endowed with the $\ell ^1$-metric

Banakh, Iryna; Банах, Тарас; Garbulińska-Węgrzyn, Joanna

doi:10.4064/cm8226-11-2020

Cited by 2 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2.The second statement follows from the definition of the distance ďF X and Theorem 25.1(4).3. The inequality ďp F X ≤ d p F X follows from the definition of the distance ďF X and Theorem 25.1(5).Next, we prove thatd ∞ F X ≤ 2n • |F n| • ď∞ F X where n = max{1, deg(F )}. To derive a contradiction, assume that d ∞ F X (a, b) > 2n • |F n| • ď∞ F X (a, b) for some elements a, b ∈ F X.…”

mentioning

confidence: 78%

“…By Proposition 5.15, there exists a non-expanding map r : EX → X such that r • e X is the identity map of X. By Theorem 25.1 (5), the maps F r : F EX → F X and F e X : F X → F EX are non-expanding. Since F r • F e X is the identity map of F X, the non-expanding map F e X : F X → F EX is an isometry and hence ďp 5.…”

Section: ) ďPmentioning

confidence: 96%

“…For any complete metric space X the completion of the metric space H ∞ X can be identified with the space of all nonempty compact subsets of X, endowed with the Hausdorff metric. On the other hand, the completion of the metric space H 1 X can be identified with the space of all nonempty compact subsets of zero length in X, see [5].…”

Section: For An Infinite Subset ω Of ω We Put [ω]mentioning

confidence: 99%

See 2 more Smart Citations

The $\ell^p$-metrization of functors with finite supports

Banakh,

Brydun,

Karchevska

et al. 2020

Preprint

View full text Add to dashboard Cite

Let p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX ), we introduce the distance d p F X on the functor-space F X as the largest distance such that for every n ∈ N and a ∈ F n the map X n → F X, f → F f (a), is non-expanding with respect to the ℓ p -metric d p X n on X n . We prove that the distance d p F X is a pseudometric if and only if the functor F preserves singletons; d p F X is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX ) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX ) → (Y, dY ) between metric spaces the map FIf the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim F p X ≤ deg(F ) • dim X. Using injective envelopes, we introduce a modification ďp F X of the distance d p F X and prove that the functor F p : Dist → Dist, F p : (X, dX ) → (F X, ďp F X ), in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces.