Comptes Rendus
no. 9
Logic/Topology
Restricting uniformly open surjections

Presented by: the Editorial Board

Tomasz Kania1, 2; Martin Rmoutil1, 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.

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→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.

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→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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.09.008

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 
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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/

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

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