T3 and T4-Objects at p in the Category of Cauchy Spaces

Year 2016, Volume: 4 Issue: 1, 6 - 14, 15.04.2016

Muammer Kula

https://doi.org/10.36753/mathenot.421346

Abstract

There are various generalization of the usual topological T3 and T4- axioms to topological categories

defined in [2] and [8]. [8] is shown that they lead to different T3 and T4 concepts, in general. In this

paper, an explicit characterizations of each of the separation properties T3 and T4 at a point p and the

generalized separation properties is given in the topological category of Cauchy spaces. Moreover,

specific relationships that arise among the various Ti, i = 0; 1; 2; 3; 4, PreT2; and T2 structures at p and the

generalized separation properties are examined in this category. Finally, we investigate the relationships

between the generalized separation properties and the separation properties at a point p in this category.

Keywords

Topological category, Cauchy space, Cauchy map, Seperation

References

• [1] Adámek, J., Herrlich, H. and Strecker, G. E., Abstract and Concrete Categories. Wiley, New York, 1990.
• [2] Baran, M., Separation properties. Indian J. Pure Appl. Math. 23 (1992), 333-341.
• [3] Baran, M., Separation Properties at p for the topological categories of reflexive relation spaces and preordered spaces. Math. Balkanica 3 (1992), 193-198.
• [4] Baran, M., The notion of closedness in topological categories. Comment. Math. Univ. Carolinae 34 (1993), 383-395.
• [5] Baran, M., Generalized local separation properties. Indian J. Pure Appl. Math. 25 (1994), 615-620.
• [6] Baran, M., Separation properties in topological categories. Math. Balkanica 10 (1996), 39-48.
• [7] Baran, M., Completely regular objects and normal objects in topological categories. Acta Math. Hungar. 80 (1998), no. 3, 211-224.
• [8] Baran, M., T3 and T4-objects in topological categories. Indian J. Pure Appl. Math. 29 (1998), 59-69.
• [9] Baran, M., Closure operators in convergence spaces. Acta Math.Hungar. 87 (2000), 33-45.
• [10] Baran, M., Compactness, perfectness, separation, minimality and closedness with respect to closure operators. Applied Categorical Structures 10 (2002), 403-415.
• [11] Baran, M., PreT 2-objects in topological categories. Applied Categorical Structures 17 (2009), 591-602.
• [12] Baran, M. and Al-Safar, J., Quotient-reflective and bireflective subcategories of the category of preordered sets. Topology and its Appl. 158 (2011), 2076-2084.
• [13] Baran, M. and Altındi¸s, H., T0-Objects in topological categories. J. Univ. Kuwait (sci.) 22 (1995), 123-127.
• [14] Baran, M. and Altındi¸s, H., T2-objects in topological categories. Acta Math. Hungar. 71 (1996), no. 1-2, 41-48.
• [15] Dikranjan, D. and Giuli, E., Closure operators I. Topology Appl. 27 (1987), 129-143.
• [16] Johnstone, P. T., Topos Theory. London Math. Soc. Monographs, No.10 Academic Press, New York, 1977.
• [17] Katetov, M., On continuity structures and spaces of mappings. Comm. Math. Univ. Car. 6 (1965), 257-278.
• [18] Keller, H., Die limes-uniformisierbarkeit der limesräume. Math. Ann. 176 (1968), 334-341.
• [19] Kowalsky, H. J., Limesräume und komplettierung. Math. Nachr. 12 (1954), 301-340.
• [20] Kula, M., A note on Cauchy spaces. Acta Math. Hungar. 133 (2011), no. 1-2, 14-32.
• [21] Kula, M., Separation properties at p for the topological category of Cauchy Spaces. Acta Math. Hungar. 136 (2012), no. 1-2, 1-15.
• [22] Lowen-Colebunders, E., Function classes of Cauchy Continuous maps. M. Dekker, New York, 1989.
• [23] Marny, Th., Rechts-Bikategoriestrukturen in Topologischen Kategorien. Dissertation, Freie Universität Berlin, 1973.
• [24] Mielke, M. V., Separation axioms and geometric realizations. Indian J. Pure Appl. Math. 25 (1994), 711-722.
• [25] Nel, L. D., Initially structured categories and cartesian closedness. Canad. Journal of Math. XXVII (1975), 1361- 1377.
• [26] Preuss, G., Theory of Topological Structures. An Approach to Topological Categories. D. Reidel Publ. Co., Dordrecht, 1988.
• [27] Preuss, G., Improvement of Cauchy spaces. Q&A in General Topology 9 (1991), 159-166.
• [28] Preuss, G., Semiuniform convergence spaces. Math. Japon. 41 (1995), 465-491.
• [29] Rath, N., Precauchy spaces. PH.D. Thesis, Washington State University, 1994.
• [30] Rath, N., Completion of a Cauchy space without the T2-restriction on the space. Int. J. Math. Math. Sci. 3, 24 (2000), 163-172.
• [31] Weck-Schwarz, S., T0-objects and separated objects in topological categories. Quaestiones Math. 14 (1991), 315-325.