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Year 2021, Volume: 50 Issue: 3, 612 - 623, 07.06.2021
https://doi.org/10.15672/hujms.740593

Abstract

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.

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

Year 2021, Volume: 50 Issue: 3, 612 - 623, 07.06.2021
https://doi.org/10.15672/hujms.740593

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.

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.
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Muhammad Qasim This is me 0000-0001-9485-8072

Bin Pang 0000-0001-5092-8278

Publication Date June 7, 2021
Published in Issue Year 2021 Volume: 50 Issue: 3

Cite

APA Qasim, M., & Pang, B. (2021). Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces. Hacettepe Journal of Mathematics and Statistics, 50(3), 612-623. https://doi.org/10.15672/hujms.740593
AMA Qasim M, Pang B. Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces. Hacettepe Journal of Mathematics and Statistics. June 2021;50(3):612-623. doi:10.15672/hujms.740593
Chicago Qasim, Muhammad, and Bin Pang. “Pre-Hausdorff and Hausdorff Objects in the Category of Quantale-Valued Closure Spaces”. Hacettepe Journal of Mathematics and Statistics 50, no. 3 (June 2021): 612-23. https://doi.org/10.15672/hujms.740593.
EndNote Qasim M, Pang B (June 1, 2021) Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces. Hacettepe Journal of Mathematics and Statistics 50 3 612–623.
IEEE M. Qasim and B. Pang, “Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 612–623, 2021, doi: 10.15672/hujms.740593.
ISNAD Qasim, Muhammad - Pang, Bin. “Pre-Hausdorff and Hausdorff Objects in the Category of Quantale-Valued Closure Spaces”. Hacettepe Journal of Mathematics and Statistics 50/3 (June 2021), 612-623. https://doi.org/10.15672/hujms.740593.
JAMA Qasim M, Pang B. Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces. Hacettepe Journal of Mathematics and Statistics. 2021;50:612–623.
MLA Qasim, Muhammad and Bin Pang. “Pre-Hausdorff and Hausdorff Objects in the Category of Quantale-Valued Closure Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, 2021, pp. 612-23, doi:10.15672/hujms.740593.
Vancouver Qasim M, Pang B. Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces. Hacettepe Journal of Mathematics and Statistics. 2021;50(3):612-23.