S. A. Bogatyi | Searcharxiv

arXiv2025

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 $δ$-surjective $d$-isometry between such equicardinal cardinality homogeneous metric spaces $X$ and $Y$, then there exists a bijective $(d+2δ)$-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.

math.MG↗

arXiv2016

On Topologies on the Group $(\Z_p)^{\N}$

It is proved that, on any Abelian group of infinite cardinality ${\bf m}$, there exist precisely $2^{2^{\bf m}}$ nonequivalent bounded Hausdorff group topologies. Under the continuum hypothesis, the number of nonequivalent compact and locally compact Hausdorff group topologies on the group $(\Z_p)^{\N}$ is determined.

math.GR↗