SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 5

Maps that preserve left (right) $K$-Cauchy sequences

Olivier Olela Otafudu(North-West University)0000-0001-9593-7899
Pages
—
Publication Date
2021-10-15
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Moreover, we characterize totally bounded sets on a quasi-pseudometric space in terms of maps that preserve left $K$-Cauchy and right $K$-Cauchy sequences and uniformly locally semi-Lipschitz maps.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:It is well-known that on quasi-pseudometric space $(X,q)$, every $q^s$-Cauchy sequence is left (or right) $K$-Cauchy sequence but the converse does not hold in general.
📊
Key Findings & Data:In this article, we study a class of maps that preserve left (right) $K$-Cauchy sequences that we call left (right) $K$-Cauchy sequentially-regular maps.
💡
Conclusion & Impact:Moreover, we characterize totally bounded sets on a quasi-pseudometric space in terms of maps that preserve left $K$-Cauchy and right $K$-Cauchy sequences and uniformly locally semi-Lipschitz maps.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Cauchy sequential regularity · left $K$-Cauchy · bornology

Abstract

It is well-known that on quasi-pseudometric space $(X,q)$, every $q^s$-Cauchy sequence is left (or right) $K$-Cauchy sequence but the converse does not hold in general. In this article, we study a class of maps that preserve left (right) $K$-Cauchy sequences that we call left (right) $K$-Cauchy sequentially-regular maps. Moreover, we characterize totally bounded sets on a quasi-pseudometric space in terms of maps that preserve left $K$-Cauchy and right $K$-Cauchy sequences and uniformly locally semi-Lipschitz maps.

🔖

Cite This Article

Otafudu, O. O. (2021). Maps that preserve left (right) $K$-Cauchy sequences. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 5. https://doi.org/10.15672/hujms.815689

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (16)

  1. G. Beer and M.I. Garrido, Bornologies and locally Lipschitz functions, Bull. Aust. Math. Soc. 90, 257–263, 2014.
  2. G. Beer and S. Levi, Total boundedness and bornologies, Topology Appl. 156, 1271– 1288, 2009.
  3. G. Beer and S. Levi, Strong uniform continuity, J. Math. Anal. Appl. 350, 568–589, 2009.
  4. S. Cobzas, Completeness in Quasi-Pseudometric SpacesA Survey, Mathematics, 8 (8), 1279, 2020.
  5. S. Cobzas, Functional analysis in asymmetric normed spaces, Frontiers in Mathemat- ics, Springers, Basel, 2013.
  6. G. Di Maio, E. Meccariello and S. Naimpally, Decompositions of UC spaces, Questions Answers Gen. Topology, 22, 13–22, 2004.
  7. D. Doitchinov, On completeness in quasi-metric spaces, Topology Appl. 30, 127–148, 1988.
  8. T. Jain and S. Kundu, Atsuji completions: Equivalent characterisations, Topology Appl. 154, 28–38, 2007.
  9. H-P.A. Künzi, An introduction to quasi-uniform spaces, Contemp. Math. 486, 239– 304, 2009.
  10. S.P. Moshokoa, On classes of maps with the extension property to the bicompletion in quasi-pseudo metric spaces, Quaest. Math. 28, 391–400, 2005.
  11. O. Olela Otafudu, W. Toko and D. Mukonda, On bornology of extended quasi-metric spaces, Hacet. J. Math. Stat. 48, 1767–1777, 2019.
  12. R.F. Snipes, Functions that preserve Cauchy sequences, Nieuw Arch. Voor Wiskd. 25, 409–422, 1977.
  13. I.L. Reilly, P.V. Subrahmanyam and M.K. Vamanamurthy, Cauchy sequences in quasi-pseudometric spaces, Monatsh. Math. 93, 127–140, 1982.
  14. S. Romaguera and M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98, 311–322, 1999.
  15. S. Romaguera and P. Tirado, A characterization of quasi-metric completeness in terms of -contractive mappings having fixed points, Mathematics, 8 (1), 16, 2020.
  16. T. Vroegrijk, Pointwise bornological space, Topology Appl. 156, 2019–2027, 2009.