  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.
