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Restricting uniformly open surjections
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Kania, Tomasz and Rmoutil, Martin (2017) Restricting uniformly open surjections. Comptes Rendus Mathematique, 355 (9). pp. 925-928. doi:10.1016/j.crma.2017.09.008 ISSN 1631073X.
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Official URL: https://doi.org/10.1016/j.crma.2017.09.008
Abstract
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.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Metric spaces, Vector spaces | ||||||||
| Journal or Publication Title: | Comptes Rendus Mathematique | ||||||||
| Publisher: | Elsevier Masson | ||||||||
| ISSN: | 1631073X | ||||||||
| Official Date: | September 2017 | ||||||||
| Dates: |
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| Volume: | 355 | ||||||||
| Number: | 9 | ||||||||
| Page Range: | pp. 925-928 | ||||||||
| DOI: | 10.1016/j.crma.2017.09.008 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 13 September 2017 | ||||||||
| Date of first compliant Open Access: | 19 September 2018 | ||||||||
| RIOXX Funder/Project Grant: |
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