{
  "abstract": "An elementary space is a pair $(X, \\mathcal A)$, where $\\mathcal A$ is an algebra of subsets of $X$. Every elementary space $(X, \\mathcal A)$ is naturally a topological space when equipped with the topology $τ_{\\mathcal A}$ generated by $\\mathcal A$. Elementary mappings on elementary spaces are defined in a manner analogous to measurable mappings on measurable spaces. In addition, the notion of an {\\it extension} of an elementary space is introduced. Let $M(X, \\mathcal A)$, denote the ring of all real-valued elementary functions on $(X, \\mathcal A)$. In this article, we study the ring $M(X, \\mathcal A)$ and extensions of an elementary space. We show that, for any elementary space $(X, \\mathcal A)$, there exists a unique compact elementary extension $(β_{\\mathcal A}X, \\bar{\\mathcal A})$ of $X$, analogous to the Stone-$\\check C$ech compactification, which contains every other extension of $X$. This extension has the property that every elementary map from $X$ into any compact topological space extends continuously to $β_{\\mathcal A}X$. Next, we show that $β_{\\mathcal A}X$ contains a realcompact extension $(\\upsilon_{\\mathcal A}X, \\bar{\\mathcal A}_R)$ of $X$, such that every elementary map from $X$ into any realcompact topological space extends continuously to $\\upsilon_{\\mathcal A}X$. Moreover, $\\upsilon_{\\mathcal A}X$ is the smallest realcompact extension of $X$ and $(X, \\mathcal A)$ is realcompact if and only if $X=\\upsilon_{\\mathcal A}X$. Also, we see that, every metric space equipped with the $σ$-algebra of all Borel sets is realcompact. Finally, we observe that the maximal ring of quotients of $M(X, \\mathcal A)$ is the direct limit of locally constant (continuous) functions on dense open subsets of $(X, τ_{\\mathcal A})$. Some questions raised in the literature are also answered.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15593",
  "author_total": 2,
  "authors": [
    {
      "name": "Nidaa Hasan Haji"
    },
    {
      "name": "Abdolaziz Hesari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jy4ayvngbqyw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15593",
  "fulltext_url": null,
  "id": "8jy4ayvngbqyw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15593",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221767678",
  "pdf_url": "https://arxiv.org/pdf/2610.15593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jy4ayvngbqyw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Topological extensions of elementary measurable spaces",
  "title_text": "Topological extensions of elementary measurable spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8jy4ayvngbqyw",
  "venue": "arXiv",
  "versions": []
}