Surjective Mappings in the Hyers--Ulam Theorem and the Gromov--Hausdorff Distance
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908734966464512 |
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| author | Bogatyi, S. A. Reznichenko, E. A. Tuzhilin, A. A. |
| author_facet | Bogatyi, S. A. Reznichenko, E. A. Tuzhilin, A. A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22776 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Surjective Mappings in the Hyers--Ulam Theorem and the Gromov--Hausdorff Distance Bogatyi, S. A. Reznichenko, E. A. Tuzhilin, A. A. Metric Geometry 51F99 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. |
| title | Surjective Mappings in the Hyers--Ulam Theorem and the Gromov--Hausdorff Distance |
| topic | Metric Geometry 51F99 |
| url | https://arxiv.org/abs/2512.22776 |