Restricting uniformly open surjections

Tomasz Kania; Martin Rmoutil

doi:10.1016/j.crma.2017.09.008

Logic/Topology

Restricting uniformly open surjections
[Restrictions des surjections uniformément ouvertes]

Présenté par : the Editorial Board

Tomasz Kania ^1,² ; Martin Rmoutil ^1,³

¹ Mathematics Institute, University of Warwick, Gibbet Hill Rd, Coventry, CV4 7AL, United Kingdom
² Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
³ Department of Mathematics Education, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic

Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 925-928.

Résumé
Abstract

Nous utilisons la théorie des sous-modèles élémentaires pour améliorer un résultat récent d'Aron, Jaramillo et Le Donne (2017) [1] sur les restrictions de surjections continues, uniformément ouvertes, à des sous-espaces où elles restent surjectives. Précisément, supposons que X et Y sont des espaces métriques et $f : X \to Y$ une surjection continue. Si X est complet et f est uniformément ouverte, alors X contient un sous-espace fermé Z de même densité que Y, tel que la restriction de f à Z est encore uniformément ouverte et surjective. De plus, si X est un espace de Banach, alors Z peut être pris sous-espace linéaire fermé. La contrepartie de ce théorème pour les espaces uniformes est aussi démontrée.

We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (2017) [1] concerning restricting uniformly open, continuous surjections to smaller subspaces where they remain surjective. To wit, suppose that X and Y are metric spaces and let $f : X \to Y$ be a continuous surjection. If X is complete and f is uniformly open, then X contains a closed subspace Z with the same density as Y such that f restricted to Z is still uniformly open and surjective. Moreover, if X is a Banach space, then Z may be taken to be a closed linear subspace. A counterpart of this theorem for uniform spaces is also established.

Reçu le : 2017-07-10
Accepté le : 2017-09-12
Publié le : 2017-09-01

DOI : 10.1016/j.crma.2017.09.008

Affiliations des auteurs :

Tomasz Kania ^{1, 2} ; Martin Rmoutil ^{1, 3}

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Tomasz Kania; Martin Rmoutil. Restricting uniformly open surjections. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 925-928. doi : 10.1016/j.crma.2017.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.008/

Bibliographie
Cité par

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Cité par Sources :

^☆ The authors acknowledge with thanks funding received from the European Research Council; ERC Grant Agreement No. 291497.

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