Abstract
The densifiability of a metric space (X, d) is the susceptibility of being filled by Peano continua. In the present paper we introduce the notion of densifiability on the hyperspace obtained from a determined collection of subsets of a metric space (X, d) endowed with a metric structure. Concretely, by using two theorems of Borsuk-Mazurkiewicz and Michael, we show that if the space (X, d) is a continuum, then the hyperspace \({{\mathcal {C}}}{{\mathcal {L}}}(X)\) of all closed non-empty subsets of (X, d) is densifiable for the Hausdorff metric \({\mathcal {H}}_{d}\). Likewise, we prove that the densifiability of the hyperspace \(({{\mathcal {C}}}{{\mathcal {L}}}(X),{\mathcal {H}}_{d})\) implies that any completion of (X, d) is a continuum provided that (X, d) is bounded. From this result we give a characterization of the densifiability of \(({{\mathcal {C}}}{{\mathcal {L}}}(X),{\mathcal {H}}_{d})\) for a bounded and complete metric space (X, d). Finally, from old Bing, Moise and Menger theorems, we prove that any dense subspace of a Peano continuum has a densifiable hyperspace.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht (1993)
Bing, R.H.: Extending a metric. Duke Math. J. 14, 511–519 (1947)
Bing, R.H.: Partitioning a set. Bull. Am. Math. Soc. 55(12), 1101–1110 (1949)
Borsuk, K., Mazurkiewicz, S.: Sur l’hyperespace d’un continu. C. R. Soc. Sc. Varsovie 24, 149–152 (1931)
Costantini, C., Kubis, W.: Paths in hyperspaces. Appl. Gen. Topol. Univ. Politécnica Valencia 4(2), 377–390 (2003)
Caballero, J., Rocha, J., Sadarangani, K.: Existence of a fractal of iterated function systems containing condensing functions for the degree of nondensifiability. J. Fixed Point Theory Appl. 25, 1 (2023). https://doi.org/10.1007/s11784-022-00995-0
Dugundji, J.: Topology. Allyn and Bacon Inc, Boston (1966)
García, G.: The minimal displacement problem of DND-Lipschitzian mappings. Bol. Soc. Mat. Mex. 29, 38 (2023). https://doi.org/10.1007/s40590-023-00512-4
García, G.: Continuous global optimization on fractals through \(\alpha \)-dense curves. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117, 165 (2023). https://doi.org/10.1007/s13398-023-01493-9
Garcia, G., Mora, G.: The degree of convex nondensifiability in Banach spaces. J. Convex Anal. 22, 871–888 (2015)
Garcia, G., Mora, G.: A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral equations. J. Math. Anal. Appl. 472(1), 1220–1235 (2019)
García, G., Mora, G.: Iterated function systems based on the degree of nondensifiability. J. Fractal Geom. 9(3/4), 357–372 (2022). https://doi.org/10.4171/JFG/121
Hausdorff, F.: Mengenlehre, 3d edn. Springer, Berlin (1927)
Hocking, J.G., Young, G.S.: Topology. Dover Publications Inc, New York (1988)
Illanes, A., Nadler, S.B.: Hyperspaces: Fundamental and Recent Advances. Marcel Dekker Inc, New York (1999)
Kelley, J.L.: Hyperspaces of a continuum. Trans. Am. Math. Soc. 52, 23–36 (1942)
Kelley, J.L.: General Topology. D. van Norstrand Company Inc, New York (1955)
Mazurkiewicz, S.: Sur l’hyperespace d’un continu. Fundam. Math. 18, 171–177 (1932)
Menger, K.: Untersuchungen über allgemeine Metrik I, II, III. Math. Ann. 100, 75–163 (1928)
Michael, E.: Topologies on spaces of subsets. Trans. Am. Math. Soc. 71, 152–182 (1951)
Moise, E.E.: Grille decomposition and convexification theorems for compact locally connected continua. Bull. Am. Math. Soc. 55, 1111–1121 (1949)
Mora, G.: The Peano curves as limit of \(\alpha \)-dense curves. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 9, 23–28 (2005)
Mora, G.: Some density properties of the closed unit ball of L1. Topol. Appl. 156, 2246–2256 (2009)
Mora, G., Cherruault, Y.: Characterization and generation of \(\alpha \)-dense curves. Comput. Math. Appl. 33, 83–91 (1997)
Mora, G., Benavent, R., Navarro, J.C.: Polynomial alpha-dense curves and multiple integration. Int. J. Comput. Numer. Anal. Appl. 1(1), 55–68 (2002)
Mora, G., Mira, J.A.: Alpha-dense curves in infinite dimensional spaces. Int. J. Pure Appl. Math. 5(4), 437–449 (2003)
Mora, G., Redtwitz, D.A.: Densifiable metric spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 105(1), 71–83 (2011)
Nadler, S.: Hyperspaces of Sets. Marcel-Dekker, New York (1978)
Sagan, H.: Space-filling Curves. Springer, New York (1994)
Toranzos, F.A.: Embedding of convex metric spaces in Euclidean spaces (In Spanish). Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires (1966). http://digital.bl.fcen.uba.ar/Download/Tesis/Tesis_1279_Toranzos.pdf
Vietoris, L.: Bereiche Zweiter Ordnung. Monatsh. Math. Phys. 33, 49–62 (1923)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
López-Pezoa, E., Mora, G. & Redtwitz, D.A. Densifiability in hyperspaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 55 (2024). https://doi.org/10.1007/s13398-024-01557-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-024-01557-4