2025/12/28 by Bogatyi, S. A., Reznichenko, E. A., Tuzhilin, A. A.
#51F99 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2512.22776
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.