The notions of closedness and D-connectedness in quantale-valued approach spaces

Document Type : Research Paper


1 Department of Mathematics, School of Natural Sciences, National University of Sciences & Technology, Islamabad.

2 Department of Mathematics, Hacı Bektaş Veli University, Nevşehir, Turkey



In this paper, we characterize local $T_{0}$ and $T_{1}$ quantale-valued gauge spaces, show how these concepts are related to each other and apply them to $\mathcal{L}$-approach distance spaces and $\mathcal{L}$-approach system spaces. Furthermore, we give the characterization of a closed point and $D$-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.


[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories", John Wiley and Sons, 1990.
[2] Baran, M., Separation properties, Indian J. Pure Appl. Math. 23 (1991), 333-341.
[3] Baran, M., The notion of closedness in topological categories, Comment. Math. Univ. Carolin. 34(2) (1993), 383-395.
[4] Baran, M. and Altındis, H., T2 objects in topological categories, Acta Math. Hungar. 71(1-2) (1996), 41-48.
[5] Baran, M., Separation properties in topological categories, Math. Balkanica (N.S.) 10 (1996), 39-48.
[6] Baran, M., T3 and T4-objects in topological categories, Indian J. Pure Appl. Math. 29 (1998), 59-70.
[7] Baran, M., Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80(3) (1998), 211-224.
[8] Baran, M., Closure operators in convergence spaces, Acta Math. Hungar. 87(1-2) (2000), 33-45.
[9] Baran, M., Compactness, perfectness, separation, minimality and closedness with respect to closure operators, Appl. Categ. Structures 10(4) (2002), 403-415.
[10] Baran, M. and Al-Safar, J., Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology Appl. 158(15) (2011), 2076-2084.
[11] Baran, M., Kula, S., Baran, T.M., and Qasim, M., Closure operators in semiuniform convergence spaces, Filomat 30(1) (2016), 131-140.
[12] Baran, M. and Qasim, M., Local T0 approach spaces, Math. Sci. Appl. E-Notes 5(1) (2017), 46-56.
[13] Baran, M. and Qasim, M., T1 Approach spaces, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 68(1) (2019), 784-800.
[14] Baran, T.M., "T0 and T1 Pseudo-quasi-semi Metric Space", Ph.D. Thesis, Erciyes University, 2018.
[15] Berckmoes, B., Lowen, R., and Van Casteren, J., Approach theory meets probability theory, Topology Appl. 158(7) (2011), 836-852.
[16] Brock, P. and Kent, D., On convergence approach spaces, Appl. Categ. Structures 6(1) (1998), 117-125.
[17] Colebunders, E., De Wachter S., and Lowen R., Intrinsic approach spaces on domains, Topology Appl. 158(17) (2011), 2343-2355.
[18] Colebunders, E., De Wachter, S., and Lowen, R., Fixed points of contractive maps on dcpo’s, Math. Structures Comput. Sci. 24(1) (2014), 1-18.
[19] Dikranjan, D. and Giuli, E., Closure operators I, Topology Appl. 27(2) (1987), 129- 143.
[20] Flagg, R.C., Quantales and continuity spaces, Algebra Universalis 37 (1997), 257- 276.
[21] Höhle, U., Commutative, residuated l-monoids, In: Non-classical logics and their applications to fuzzy subsets, Springer 32 (1995), 53-106.
[22] Jäger, G., Probabilistic approach spaces, Math. Bohem. 142(3) (2017), 277-298.
[23] Jäger, G. and Yao, W., Quantale-valued gauge spaces, Iran. J. Fuzzy Syst. 15(1) (2018), 103-122.
[24] Jäger, G., Quantale-valued generalization of approach spaces: L-approach systems, Topology Proc. 51 (2018), 253-276.
[25] Jäger, G., Quantale-valued generalizations of approach spaces and quantale-valued topological spaces, Quaest. Math. (2018, online),
[26] Klement, E.P., Mesiar, R., and Pap, E., "Triangular Norms", Springer, 2000.
[27] Kula, M., A note on Cauchy spaces, Acta Math. Hungar. 133(1-2) (2011), 14-32.
[28] Kula, M., Maraslı, T., and Özkan, S., A note on closedness and connectedness in the category of proximity spaces, Filomat 28(7) (2014), 1483-1492.
[29] Lai, H. and Tholen, W., Quantale-valued approach spaces via closure and convergence,
[30] Lowen, R., Approach spaces A common supercategory of TOP and MET, Math. Nachr. 141(1) (1989), 183-226.
[31] Lowen, R., "Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad", Oxford University Press, 1997.
[32] Preuss, G., "Theory of Topological Structures: An Approach to Categorical Topology", D. Reidel Publ. Co., 1988.
[33] Preuss, G., "Foundations of Topology: An Approach to Convenient Topology", Kluwer Academic Publishers, 2002.
[34] Schweizer, B. and Sklar, A., "Probabilistic Metric Spaces", North Holland, 1983