SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2012 / Cilt: 41 - Sayı: 1

A short note on the role of grills in nearness frames

Martin M. MUGOCHI
Pages
85–91
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)

Some of the results we establish here are that, in a separated Boolean nearness frame, a near grill is contained in a unique cluster, and that, in quotient-fine nearness frames with spatial completion, near subsets are contained in near grills.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Grills and clusters have featured quite significantly in the theory of nearness spaces (see for example H.
📊
Key Findings & Data:Herrlich (Topological structures, Topological Structures 1.
💡
Conclusion & Impact:Some of the results we establish here are that, in a separated Boolean nearness frame, a near grill is contained in a unique cluster, and that, in quotient-fine nearness frames with spatial completion, near subsets are contained in near grills.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

Grills and clusters have featured quite significantly in the theory of nearness spaces (see for example H. Herrlich (Topological structures, Topological Structures 1. Math. Centre Tracts, 59–122, 1974) and G. Preuss, (Theory of topological structures: An approach to categorical topology (D. Reidel Pub. Co., 1987)). In this paper we consider the role they play in nearness frames. Some of the results we establish here are that, in a separated Boolean nearness frame, a near grill is contained in a unique cluster, and that, in quotient-fine nearness frames with spatial completion, near subsets are contained in near grills.

🔖

Cite This Article

MUGOCHI, M. M. (2022). A short note on the role of grills in nearness frames. Hacettepe Journal of Mathematics and Statistics, Cilt: 41 - Sayı: 1, 85–91. https://ajindex.com/articles/530481

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (12)

  1. Banaschewski, B. Uniform completion in pointfree topology (Topological and algebraic structures in fuzzy sets 19-56, Trends Log. Stud. Log. Libr., 20, Kluwer Acad. Publ., Dordrecht, 2003)
  2. Banaschewski, B., Hong, S. S. and Pultr, A. On the completion of nearness frames, Quaest. Math. 28, 19-37, 1998
  3. Banaschewski, B. and Pultr, A. Cauchy points of uniform and nearness frames, Quaest. Math. 19, 107-127, 1996
  4. Bentley, H. L. and Herrlich, H. Extensions of topological spaces, In: Topology, Proceedings of the Memphis State University Conference, 1975, Ed.: S.P. Franklin and B.V. Smith-Thomas, Marcel Dekker, New York, 129-184, 1976
  5. Choquet, G. Sur les notions de filtre et de grille, Comptes Rendus Acad. Sci. Paris 224, 171-173, 1947
  6. Dube, T. Structures in frames (PhD Thesis, University of Durban-Westville, 1992)
  7. Gratzer, G. Lattice Theory: First concepts and distributive lattices (W. H. Freeman and Co., San Fransisco, Carlifornia, 1971)
  8. Herrlich, H. Topological structures, Topological Structures 1, Math. Centre Tracts, 59-122, 1974
  9. Johnstone, P. T. Stone Spaces (Cambridge Univ. Press, Cambridge, 1982)
  10. Mugochi, M. M. Contributions to the theory of nearness in pointfree topology (PhD Thesis, University of South Africa, 2009)
  11. Preuss, G. Theory of topological structures: An approach to categorical topology (D. Reidel Pub Co., 1987)
  12. Thron, W. J. Proximity structures and grills, Math. Ann. 206, 35-62, 1973.