SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2019 / Cilt: 48 - Sayı: 6

Bornological quasi-metrizability in generalized topology

Artur PIEKOSZ
Eliza WAJCH
Pages
1653–1666
Publication Date
2022-10-18
DOI
—
🏛️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)

Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced.
📊
Key Findings & Data:Quasi-metrization theorems for generalized bornological universes are deduced.
💡
Conclusion & Impact:Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Delfs-Knebusch generalized topological space · quasi-metric · bornology

Abstract

A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced. Quasi-metrization theorems for generalized bornological universes are deduced. A uniform quasi-metrizability with respect to a bornology is studied. The class of locally small spaces is considered and a possibly larger class of weakly locally small spaces is defined. The proofs and numerous examples are given in ZF. An example of a weakly locally small space which is not locally small is constructed under ZF+CC. Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.

🔖

Cite This Article

PIEKOSZ, A., WAJCH, E. (2022). Bornological quasi-metrizability in generalized topology. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 6, 1653–1666. https://ajindex.com/articles/529761

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (22)

  1. J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, the Joy of Cats, Wiley, New York, 1990
  2. A. Appert and K. Fan, Espaces Topologiques Intermédiaires, Act. Sci. et Ind. 1121, Hermann, Paris, 1951
  3. E. Colebunders and R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (5), 1547–1556, 2004. [4] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (4), 351–357, 2002
  4. H. Delfs and M. Knebusch, Locally Semialgebraic Spaces, in: Lecture Notes in Math. 1173, Springer, Berlin-Heidelberg, 1985
  5. M.J. Edmundo and L. Prelli, Sheaves on T-topologies, J. Math. Soc. Japan, 68 (1), 347–381, 2016
  6. P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982
  7. I. Garrido and A.S. Meroño, Uniformly metrizable bornologies, J. Convex Anal. 20 (1), 285–299, 2013
  8. H. Herrlich, Axiom of Choice, Springer-Verlag, Berlin-Heidelberg, 2006
  9. D. Hofmann, G.J. Seal and W. Tholen, Monoidal Topology, A Categorical Approach to Order, Metric, and Topology, Cambridge University Press, Cambridge, 2014
  10. H. Hogbe-Nlend, Bornologies and Functional Analysis, North-Holland, Amsterdam, 1977
  11. S.T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 28, 287–320, 1949
  12. J.C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc. 13, 71–89, 1963
  13. M. Knebusch, Weakly Semialgebraic Spaces, in: Lecture Notes in Math. 1367, Springer-Verlag, New York, 1989
  14. K. Kunen, The Foundations of Mathematics, Individual Authors and College Publications, London, 2009
  15. S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, Berlin- Heidelberg, 1971
  16. A. Piękosz, On generalized topological spaces I, Ann. Polon. Math. 107 (3), 217–241, 2013
  17. A. Piękosz, On generalized topological spaces II, Ann. Polon. Math. 108 (2), 185–214, 2013
  18. A. Piękosz, O-minimal homotopy and generalized (co)homology, Rocky Mountain J. Math. 43 (2), 573–617, 2013
  19. A. Piękosz and E. Wajch, Compactness and compactifications in generalized topology, Topology Appl. 194, 241–268, 2015
  20. A. Piękosz, and E. Wajch, Quasi-Metrizability of Bornological Biuniverses in ZF, J. Convex Anal. 22 (4), 1041–1060, 2015
  21. S. Salbany, Bitopological spaces, compactifications and completions, in: Math. Monographs, Univ. of Cape Town, Cape Town, 1974
  22. T. Vroegrijk, Pointwise bornological spaces, Topology Appl. 156, 2019–2027, 2009.