ST2, ∆T2, ST3, ∆T3, Tychonoff, compact and ∂-connected objects in the category of proximity spaces

Abstract

In this paper, an explicit characterization of the separation properties $ST_2$, $\Delta T_2$, $ST_{3}$, $ \Delta T_{3} $ and Tychonoff objects are given in the topological category of proximity space. Furthermore, the (strongly) compact object and $\partial$-connected object are also characterized in the category of proximity space. Moreover, we investigate the relationships among  $ST_2$, $\Delta T_2$, $ST_{3}$, $ \Delta T_{3} $, the separation properties at a point $p$, the generalized separation properties $T_{i}$, $i=0,1,2$, $\mathbf{T_{0}}$, $\mathbf{T_{1}}$, $\mathbf{T_{2}}$ and Tychonoff objects in this category. Finally, we investigate the relationships between $\partial$-connected object and (strongly) connected object in the topological category of proximity space.

Keywords

• topological category
• proximity space
• separation
• closedness
• connectedness
• compact objects

References

1. J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories, John Wiley and Sons, New York, 1990.
2. M. Baran, Separation properties, Indian J. Pure Appl. Math. 23 (5), 333-341, 1992.
3. M. Baran, Separation Properties at p for the topological categories of reflexive relation spaces and preordered spaces, Math. Balkanica (N.S.) 6 (2), 193-198, 1992.
4. M. Baran, The notion of closedness in topological categories, Comment. Math. Univ. Carolin. 34 (2), 383-395, 1993.
5. M. Baran, Generalized local separation properties, Indian J. Pure Appl. Math. 25 (6), 615-620, 1994.
6. M. Baran, Separation properties in topological categories, Math. Balkanica (N.S.) 10 (1), 39-48, 1996.
7. M. Baran, A notion of compactness in topological categories, Publ. Math. Debrecen 50 (3-4), 221-234, 1997.
8. M. Baran, $T_3$ and $T_4$-objects in topological categories, Indian J. Pure Appl. Math. 29 (1), 59-69, 1998.

9. M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80 (3), 211-224, 1998.
10. M. Baran, Closure operators in convergence spaces, Acta Math. Hungar. 87 (1-2), 33-45, 2000.
11. M. Baran, Compactness, perfectness, separation, minimality and closedness with re- spect to closure operators, Appl. Categ. Structures 10 (4), 403-415, 2002.
12. M. Baran, Pre$T_2$-objects in topological categories, Appl. Categ. Structures 17 (6), 591-602, 2009, DOI: 10.1007/s10485-008-9161-4.
13. M. Baran and J. Al-Safar, Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology Appl. 158 (15), 2076-2084, 2011, DOI 10.1016/j.topol.2011.06.043.
14. M. Baran and M. Kula, A note on separation and compactness in categories of con- vergence spaces, Proceedings of the International Conference on Applicable General Topology, Ankara, 2001, Appl. Gen. Topol. 4 (1), 1-13, 2003.
15. M. Baran and M. Kula, A note on connectedness, Publ. Math. Debrecen 68 (3-4), 489-501, 2006.
16. N. Bourbaki, General Topology, Addison-Wesley Publ. Co., 1966.
17. D. Dikranjan and E. Giuli, Closure operators I, Proceedings of the 8th international conference on categorical topology, L’Aquila, 1986, Topology Appl. 27 (2), 129-143, 1987.
18. V.A. Efremovich, Infinitesimal spaces (Russian), Doklady Akad. Nauk SSSR (N.S.) 76, 341-343, 1951.
19. L.M. Friedler, Mappings of Proximity Spaces, PH.D. Thesis, University of Alberta, 1972.
20. L.M. Friedler, Quotients of proximity spaces, Proc. Amer. Math. Soc. 37 (2), 589-594, 1973.
21. W.N. Hunsaker and P.L. Sharma, Proximity spaces and topological functors, Proc. Amer. Math. Soc. 45, 419-425, 1974.
22. P.T. Johnstone, Topos theory, London Mathematical Society Monographs, Vol. 10. Academic Press, Harcourt Brace Jovanovich, Publishers, London-New York, 1977.
23. M. Kula, A notion of connectedness in topological categories, PH.D. Thesis, University of Erciyes, 2003.
24. M. Kula, A note on Cauchy spaces, Acta Math. Hungar. 133 (1-2), 14-32, 2011, DOI: 10.1007/s10474-011-0136-9.
25. M. Kula, Separation properties at p for the topological category of Cauchy spaces, Acta Math. Hungar. 136 (1-2), 1-15, 2012, DOI: 10.1007/s10474-012-0238-z.
26. M. Kula, T. Maraşlı and S. Özkan, A note on closedness and connectedness in the category of proximity spaces, Filomat 28 (7), 1483-1492, 2014. DOI 10.2298/FIL1407483K.
27. M. Kula, S. Özkan and T. Maraşlı, Pre-Hausdorff and Hausdorff proximity spaces, Filomat 31 (12), 3837-3846, 2017.
28. S. Leader, On products of proximity spaces, Math. Ann. 154, 185-194, 1964.
29. M.W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. 15, 417-422, 1963.
30. M.V. Mielke, Separation axioms and geometric realizations, Indian J. Pure Appl. Math. 25 (7), 711-722, 1994.
31. J.R. Munkres, Topology: A First Course, Prentice-Hall, 1975.
32. S.A. Naimpally and B.D. Warrack, Proximity spaces, Cambridge Tracts in Mathe- matics and Mathematical Physics, No. 59 Cambridge University Press, London-New York, 1970.
33. W.J. Pervin, Quasi-proximities for topological spaces, Math. Ann. 150, 325-326, 1963.
34. G. Preuss, Theory of topological structures An approach to categorical topology, Math- ematics and its Applications, 39. D. Reidel Publishing Co., Dordrecht, 1988.
35. P.L. Sharma, Proximity bases and subbases, Pacific J. Math. 37, 515-526, 1971.
36. J.L. Sieber, Generalizations of Some Topological Notions to Syntopogenic Spaces, PhD. Thesis, The Pennslyvania State University, 1963.
37. Y.M. Smirnov, On completeness of proximity spaces I, Amer. Math. Soc. Transl. 38, 37-73, 1964.
38. Y.M. Smirnov, On proximity spaces, Mat. Sb. 31 (73), 543-574, 1952; Amer. Math. Soc. Transl. 38 (2), 5-35, 1964.
39. A.D. Wallace, Separation spaces, Ann. of Math. 42 (2), 687-697, 1941.
40. A.D. Wallace, Separation spaces II, Anais Acad. Brasil. Ci. 14, 203-206, 1942.
41. S. Willard, General Topology, Addison-Wesley, Reading, MA, 1970.