T₁ Approach Spaces

Abstract

In this paper, we characterize both T₁ and local T₁ limit (resp. gauge) approach spaces as well as show how these concepts are related to each other. Finally, we compare these T₁ and the usual T₁ approach spaces.

Keywords

• Topological category,T₁ objects,initial lift,limit-approach space

References

1. Adamek, J., Herrlich, H. and Strecker. G. E., Abstract and Concrete Categories, Pure and Applied Mathematics, John Wiley & Sons, New York, 1990. Baran, M., Separation properties, Indian J. pure appl. Math, 23 (1991), 333-341.
2. Baran, M., Separation Properties in Topological Categories, Math. Balkanica, 10 (1996), 39-48.
3. Baran, M., T₃ and T₄-objects in topological categories, Indian J. pure appl. Math., 29 (1998), 59-70. BaranNormal : Baran, M., Completely regular objects and normal objects in topological categories, Acta Mathematica Hungarica, 80.3 (1998), 211-224.
4. Baran, M., Closure operators in convergence spaces, Acta Mathematica Hungarica, 87.1-2 (2000), 33-45.
5. Baran, M. and Al-Safar, J., Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology and its Applications, 158.15 (2011), 2076-2084.
6. Baran, M., Kula, S. and Erciyes, A., T₀ and T₁ semiuniform convergence spaces, Filomat, 27.4 (2013), 537-546.
7. Baran, M., Kula, S., Baran, T. M. and Qasim, M., Closure Operators in Semiuniform Convergence Spaces, Filomat 30.1 (2016), 131-140.
8. Baran, M. and Qasim, M., Local T₁ Distance-Approach Spaces, Proceedings of 5th International Conference on Advanced Technology & Sciences (ICAT), Bahcesehir University, Istanbul, Turkey, (2017), 112-116.

9. Baran, T.M. and Kula, M., T₁ Extended Pseudo-Quasi-Semi Metric Spaces, Math. Sci. Appl. E-Notes, 5.1 (2017), 40-45.
10. Berckmoes, B., Lowen, R. and Van Casteren, J., Approach theory meets probability theory, Topology and its Applications, 158.7 (2011), 836-852.
11. Colebunders, E., De Wachter S., and Lowen R., Intrinsic approach spaces on domains, Topology and its Applications, 158.17 (2011), 2343-2355.
12. Dikranjan, D. and Giuli, E., Closure operators I, Topology and its Applications, 27.2 (1987), 129-143.
13. Dikranjan, D. and Tholen, W., Categorical structure of closure operators: with applications to topology, algebra and discrete mathematics, Kluwer Academic Publishers, Dordrecht, 1995.
14. Jager, G., A note on neighbourhoods for approach spaces, Hacettepe Journal of Mathematics and Statistics, 41.2 (2012), 283-290.
15. Lowen, R., Approach spaces A common Supercategory of TOP and Met, Mathematische Nachrichten, 141.1 (1989), 183-226.
16. Lowen, R., Approach spaces: The missing link in the Topology-Uniformity-Metric triad, Oxford University Press, 1997.
17. Lowen, R. and Windels, B., Approach groups, The Rocky Mountain Journal of Mathematics, 30 (2000), 1057-1073.
18. Lowen, R. and Sioen, M., A note on separation in AP, Applied general topology, 4.2 (2003), 475-486.
19. Lowen, R., and Verwulgen, S., Approach vector spaces, Houston J. Math , 30.4 (2004), 1127-1142.
20. Lowen, R., Index Analysis: Approach theory at work, Springer, 2015.
21. Preuss, G., Theory of topological structures: an approach to categorical topology, D. Reidel Publ. Co., Dordrecht, 1988.
22. Preuss, G., Foundations of topology: an approach to convenient topology, Kluwer Academic Publishers, Dordrecht, 2002.