{
  "abstract": "For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \\cite{us} proved that if $C_p(X)$ is uniformly homeomorphic to $C_p(Y)$, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. The second author and Vuma \\cite{valvu} have shown that linear homeomorphisms between $C_p^*(X)$ and $C_p^*(Y)$ also preserve pseudocompactness. Recently Baars-van Mill-Tkachuk \\cite{bmt} gave another proof of that result and raised the question if the same remains true provided $C_p^*(X)$ and $C_p^*(Y)$ are uniformly homeomorphic. In the present paper we introduce the notion of bounded uniformly continuous maps and show that every bounded uniform homeomorphism between $C_p^*(X)$ and $C_p^*(Y)$ preserve pseudocompactness. Moreover, we show that if there is a uniform homeomorphism $φ:C_p^*(X)\\to C_p^*(Y)$, where $X$ is pseudocompact and $Y$ is normal, then $Y$ is $σ$-pseudocompact, i.e. $Y$ is a countable union of closed pseudocompact subsets. It is also shown that a continuous linear map between $C_p$-spaces is norm-bounded if and only if it is bounded in our sense.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.23034",
  "author_total": 2,
  "authors": [
    {
      "name": "Mikolaj Krupski"
    },
    {
      "name": "Vesko Valov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ky3wxjhyk880.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.23034",
  "fulltext_url": null,
  "id": "7ky3wxjhyk880",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.23034",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7171502951",
  "pdf_url": "https://arxiv.org/pdf/2607.23034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ky3wxjhyk880.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2607.23034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.23034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Bounded uniform homeomorphisms between $C_p^*$-spaces preserve pseudocompactness",
  "title_text": "Bounded uniform homeomorphisms between C_p^*-spaces preserve pseudocompactness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7ky3wxjhyk880",
  "venue": "arXiv",
  "versions": []
}