Closure Operators in Constant Filter Convergence Spaces

Abstract

In this paper, we define two notions of closure in the category of constant filter convergence spaces which satisfy productivity, idempotency, and hereditariness. Moreover, by using these closure operators, we characterize each of $T_{i}$ constant filter convergence spaces, $i=0,1,2$ and show that each of these subcategories consisting of $T_{i}$ constant filter convergence spaces, $i=0,1,2$, are epireflective. Finally, we investigate the relationship among these subcategories.

Keywords

• Topological category,closure operator,constant filter convergence spaces

References

1. [1] J. Adamek, Herrlich, H., Strecker, G.E., Abstract and Concrete Categories, New York, USA, Wiley, 1990.
2. [2] M. Baran, Separation Properties, Indian J. Pure Appl. Math., 23 (1992), 333-341.
3. [3] M. Baran, The Notion of Closedness in Topological Categories, Comment. Math. Univ. Carolinae, 34 (1993), 383-395.
4. [4] M. Baran, Separation Properties in Categories of Constant Convergence Spaces, Turkish Journal of Mathematics, 18 (1994), 238-248.
5. [5] M.Baran, A Notion of Compactness in Topological Categories, Publ. Math. Debrecen, 50 (1997), 221-234.
6. [6] M. Baran, Closure Operators in Convergence Spaces, Acta Math. Hungar., 87 (2000), 33-45.
7. [7] M. Baran, Compactness, Perfectness, Separation, Minimality and Closedness with Respect to Closure Operators, Applied Categorical Structures, 10 (2002), 403-415.
8. [8] M. Baran and J. Al-Safar, Quotient-Reflective and Bireflective Subcategories of the Category of Preordered Sets, Topology and its Appl., 158 (2011), 2076-2084.

9. [9] M. Baran, Stacks and Filters, Do˘ga Mat., 16 (1992), 95-108.
10. [10] M. Baran, S. Kula, T.M. Baran and M. Qasim, Closure Operators in Semiuniform Convergence Spaces, Filomat 30 (2016), 131-140.
11. [11] M. Clementino, E. Giuli, and W. Tholen, Topology in a Category :Compactness, Port. Math., 53 (1996), 397-433.
12. [12] D. Dikranjan and E. Giuli, Closure Operators I, Topology Appl., 27 (1987), 129-143.
13. [13] D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, Kluwer Academic Publishers, Dordrecht, 1995.
14. [14] H. Herrlich, G. Salicrup and G.E. Strecker, Factorizations, Denseness, Separation, and Relatively Compact Objects, Topology Appl., 27 (1987), 157-169.
15. [15] M. Kula and M. Baran, A Note on Connectedness, Publ. Math. Debrecen, 68 (2006), 489-501.
16. [16] W. Robertson, Convergence as a Nearness Concept, Ph.D. Thesis, University of Ottawa at Carleton, 1975.
17. [17] F. Schwarz and TU. Hannover, Connections Between Convergence And Nearness, The series Lecture Notes in Mathematics, 719 (1979), 345-357.