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 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 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.
Accepted:
Published online:
Tomasz Kania 1, 2; Martin Rmoutil 1, 3
@article{CRMATH_2017__355_9_925_0, author = {Tomasz Kania and Martin Rmoutil}, title = {Restricting uniformly open surjections}, journal = {Comptes Rendus. Math\'ematique}, pages = {925--928}, publisher = {Elsevier}, volume = {355}, number = {9}, year = {2017}, doi = {10.1016/j.crma.2017.09.008}, language = {en}, }
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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