Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces

Year 2021, , 612 - 623, 07.06.2021

Muhammad Qasim Bin Pang

https://doi.org/10.15672/hujms.740593

Cited By: 3

Abstract

In previous papers, several $T_{0}$ and Hausdorff objects in topological categories are introduced and compared. The main objectives of this paper are to characterize $\overline{T_{0}}$, $T_{0}$, $T_{1}$ and pre-$\overline{T_{2}}$ objects in the category of quantale-valued closure space as well as to examine their mutual relationship.

Keywords

$\mathcal{V}$-closure space, $\mathcal{V}$-topological space, topological category, separation, Hausdorff objects

References

• [1] J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories, John Wiley & Sons, New York, 1990.
• [2] D. Aerts, Foundations of quantum physics: a general realistic and operational approach, Int. J. Theoret. Phys. 38 (1), 289–358, 1999.
• [3] D. Aerts, E. Colebunders, A. Van der Voorde and B. Van Steirteghem, State property systems and closure spaces: a study of categorical equivalence, Int. J. Theoret. Phys. 38 (1), 359–385, 1999.
• [4] G. Aumann, Kontaktrelationen, Bayer. Akad. Wiss. Math.-Nat. Kl. Sitzungsber, 67– 77, 1970.
• [5] M. Baran, Separation Properties, Indian J. Pure Appl. Math. 23, 333–341, 1991.
• [6] M. Baran, Separation properties in the categories of Constant Convergence Spaces, Turkish J. Math. 18, 238–248, 1994.
• [7] M. Baran, Separation Properties in Topological Categories, Math. Balkanica. 10, 39– 48, 1996.
• [8] M. Baran, $T_{3}$ and $T_{4}$-objects in topological categories, Indian J. Pure Appl. Math. 29, 59–70, 1998.
• [9] M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80 (3), 211–224, 1998.
• [10] M. Baran, Pre $T_{2}$ objects in topological categories, Appl. Categ. Structures, 17, 591– 602, 2009.
• [11] M. Baran and H. Altındiş, $T_{2}$ objects in topological categories, Acta Math. Hungar. 71 (1-2), 41–48, 1996.
• [12] M. Baran, D. Tokat and M. Kula, Connectedness and Separation in the Category of Closure Spaces, Filomat 24 (2), 67–79, 2010.
• [13] G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, Rhonde Island, 1940.
• [14] E. Čech, On bicompact spaces, Ann. Math. 38, 823–844, 1937.
• [15] D. Deses, E. Giuli and E. Lowen-Colebunders, On the complete objects in the category of T0 closure spaces, Appl. Gen. Topology, 4, 25–34, 2003.
• [16] M. Erné, Lattice representations for categories of closure spaces, Categorical Topology, Sigma Series in Pure Mathematics 5, Heldermann Verlag Berlin, 197–222, 1984.
• [17] R.C. Flagg, Quantales and continuity spaces, Algebra Univers. 37, 257–276, 1997.
• [18] P. Hertz, Über Axiomensysteme für beliebige Satzsysteme, Teil I, Math. Ann. 87, 246–269, 1922.
• [19] G. Jäger, Probabilistic Approach Spaces, Math. Bohem. 142 (3), 277–298, 2017.
• [20] G. Jäger and W. Yao, Quantale-valued gauge spaces, Iran. J. Fuzzy Syst. 15 (1), 103–122, 2018.
• [21] P.T. Johnstone, Stone Spaces, L. M. S. Mathematics Monograph: No. 10. Academic Press, New York, 1977.
• [22] M. Kula, A note on Cauchy spaces, Acta Math. Hungar. 133 (1-2), 14–32, 2011.
• [23] K. Kuratowski, Sur L’operation $\overline{A}$ de l’Analysis Situs, Fund. Math. 3, 182–199, 1992.
• [24] H. Lai and W. Tholen, Quantale-valued topological spaces via closure and convergence, Topology Appl. 30, 599–620, 2017.
• [25] H. Lai and W. Tholen, A note on the Topologicity of Quantale-valued Topological spaces, Log. Methods Comput. Sci. 13 (3:12), 1–13, 2017.
• [26] R. Lowen, Approach spaces: The missing link in the Topology-Uniformity-Metric triad, Oxford University Press, 1997.
• [27] M.V. Mielke, Separation axioms and geometric realizations, Indian J. Pure Appl. Math. 25, 711–722, 1994.
• [28] M.V. Mielke, Hausdorff separation and decidability, in: Symposium on Categorical Topology, University of Cape Town, Rondebosch, 155–160, 1999.
• [29] E.H. Moore, On a form of general analysis with applications to linear differential and integral equations, in: Atti del IV Congress. Internationale di Mat. II, Roma, 98–114, 1909.
• [30] B. Pang, Categorical properties of $L$-fuzzifying convergence spaces, Filomat 32 (11), 4021–4036, 2018.
• [31] B. Pang, Convenient properties of stratified $L$-convergence tower spaces, Filomat 33 (15), 4811–4825, 2019.
• [32] B. Pang, Hull operators and interval operators in the $(L,M)$-fuzzy convex spaces, Fuzzy Sets and Systems 405, 106–127, 2021.
• [33] B. Pang and F.-G. Shi, Strong inclusion orders between L-subsets and its applications in L-convex spaces, Quaest. Math. 41 (8), 1021–1043, 2018.
• [34] R.S. Pierce, Closure spaces with applications to ring theory, in: Lectures on Rings and Modules, Lecture Notes in Mathematics 246, Springer, Berlin, Heidelberg, 1972.
• [35] G. Preuss, Foundations of topology: an approach to convenient topology, Kluwer Academic Publishers, Dordrecht, 2002.
• [36] M. Qasim and S. Özkan, The notions of closedness and $D$-connectedness in Quantalevalued Approach Spaces, Categ. Gen. Alg. Struct. Appl. 12, 149–173, 2020.
• [37] G.J. Seal, Canonical and op-canonical lax algebras, Theory Appl. Categ. 14 (10), 221–243, 2005.
• [38] S. Weck-Schwarz, $T_{0}$-objects and separated objects in topological categories, Quaest. Math. 14 (3), 315–325, 1991.
• [39] Z.-Y. Xiu and Q.-H. Li, Degrees of $L$-continuity for mappings between $L$-topological spaces, Mathematics 7 (11), 1013–1028, 2019.
• [40] Z.-Y. Xiu, Q.-H. Li and B. Pang, Fuzzy convergence structures in the framework of $L$-convex spaces, Iran. J. Fuzzy Syst. 17 (4), 139–150, 2020.