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PreprintCC BY-SA 4.0arXiv

Topological extensions of elementary measurable spaces

  • Nidaa Hasan Haji
  • Abdolaziz Hesari
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Abstract

An elementary space is a pair (X, π’œ), where π’œ is an algebra of subsets of X. Every elementary space (X, π’œ) is naturally a topological space when equipped with the topology Ο„π’œ generated by π’œ. 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, π’œ), denote the ring of all real-valued elementary functions on (X, π’œ). In this article, we study the ring M(X, π’œ) and extensions of an elementary space. We show that, for any elementary space (X, π’œ), there exists a unique compact elementary extension (Ξ²π’œX, π’œΜ…) of X, analogous to the Stone-check Cech 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 Ξ²π’œX. Next, we show that Ξ²π’œX contains a realcompact extension (Ο…π’œX, π’œΜ…R) of X, such that every elementary map from X into any realcompact topological space extends continuously to Ο…π’œX. Moreover, Ο…π’œX is the smallest realcompact extension of X and (X, π’œ) is realcompact if and only if X=Ο…π’œX. Also, we see that, every metric space equipped with the Οƒ-algebra of all Borel sets is realcompact. Finally, we observe that the maximal ring of quotients of M(X, π’œ) is the direct limit of locally constant (continuous) functions on dense open subsets of (X, Ο„π’œ). Some questions raised in the literature are also answered.

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Cite

BibTeXRIS
BibTeX
@misc{haji2026topological,
  title = {{Topological extensions of elementary measurable spaces}},
  author = {Nidaa Hasan Haji and Abdolaziz Hesari},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15593},
  eprint = {2610.15593},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15593},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Topological extensions of elementary measurable spaces
AU  - Nidaa Hasan Haji
AU  - Abdolaziz Hesari
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15593
UR  - https://arxiv.org/abs/2610.15593
AB  - An elementary space is a pair (X, π’œ), where π’œ is an algebra of subsets of X. Every elementary space (X, π’œ) is naturally a topological space when equipped with the topology Ο„_π’œ generated by π’œ. 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, π’œ), denote the ring of all real-valued elementary functions on (X, π’œ). In this article, we study the ring M(X, π’œ) and extensions of an elementary space. We show that, for any elementary space (X, π’œ), there exists a unique compact elementary extension (Ξ²_π’œX, π’œΜ…) of X, analogous to the Stone-check Cech 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 Ξ²_π’œX. Next, we show that Ξ²_π’œX contains a realcompact extension (Ο…_π’œX, π’œΜ…_R) of X, such that every elementary map from X into any realcompact topological space extends continuously to Ο…_π’œX. Moreover, Ο…_π’œX is the smallest realcompact extension of X and (X, π’œ) is realcompact if and only if X=Ο…_π’œX. Also, we see that, every metric space equipped with the Οƒ-algebra of all Borel sets is realcompact. Finally, we observe that the maximal ring of quotients of M(X, π’œ) is the direct limit of locally constant (continuous) functions on dense open subsets of (X, Ο„_π’œ). Some questions raised in the literature are also answered.
ER  -