Surjective Mappings in the Hyers--Ulam Theorem and the Gromov--Hausdorff Distance

A topological space is said to be cardinality homogeneous if every nonempty open subset has the same cardinality as the space itself. Let $X$ and $Y$ be cardinality homogeneous metric spaces of the same cardinality. If there exists a $\delta$-surjective $d$-isometry between such equicardinal cardinality homogeneous metric spaces $X$ and $Y$, then there exists a bijective $(d+2\delta )$-isometry between $X$ and $Y$. This result allows us to reduce the Dilworth–Tabor theorem to the Gevirtz–Omladič–Šemrl theorem on approximation by isometries and, in particular, to questions concerning the isometry of Banach spaces.