Restricting uniformly open surjections

TL;DR

This paper uses elementary submodels to show that uniformly open surjections from complete metric spaces or Banach spaces can be restricted to smaller subspaces where they remain surjective and uniformly open, preserving key properties.

Contribution

It improves existing results by demonstrating that such restrictions can be made to smaller subspaces with the same density, including linear subspaces in Banach spaces.

Findings

Existence of closed subspaces where the surjection remains uniformly open and surjective

Extension of the result to uniform spaces

Construction of linear subspaces in Banach spaces

Abstract

We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear) 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 $f\colon X\to Y$ 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.