Research Article
Year 2025,
, 928 - 938, 24.06.2025
Abstract
In previous papers, several extensions of the $T_{2}$ separation property in topology to a topological category were compared. The aim of this paper is to develop further results relating to these extensions as well as to solve several open problems. Moreover, we show one of these $T_{2}$, namely $KT_{2}$ limit spaces and reciprocal limit spaces are equivalent and every $KT_{2}$ limit space induces the associated complete uniform limit space. Finally, we compare our results and give some applications.
Keywords
$T_{2}$ objects reciprocal limit spaces uniform limit spaces sober spaces pre-Hausdorff objects
References
- [1] J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.
- [2] M. Baran, Separation properties, Indian J. Pure Appl. Math. 23 (5), 333-341, 1991.
- [3] M. Baran, Stacks and filters, Turk. J. Math. 16, 95-108, 1992.
- [4] M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80, 211-224, 1998.
- [5] M. Baran, Pre$T_2$ Objects in topological categories, Appl. Categor. Struct. 17, 591-602, 2009.
- [6] T. M. Baran, Closedness, separation and connectedness in pseudo-quasi-semi metric spaces, Filomat, 34 (14), 4757-4766, 2020.
- [7] M. Baran, Separation, connectedness and disconnectedness, Turk. J. Math. 47, 279- 295, 2023.
- [8] M. Baran, Stone spaces I, Filomat, 38 (16), 2024.
- [9] M. Baran and H. Altindis, $T_0$-objects in topological categories, J. Univ. Kuwait 22, 123-127, 1995.
- [10] M. Baran and H. Altindis, $T_2$-objects in topological categories, Acta Math. Hungar. 71, 41-48, 1996.
- [11] M. Baran and J. Al-Safar, Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology Appl. 158, 2076-2084, 2011.
- [12] M. Baran and H. Ebughalwa, Sober spaces, Turk. J. Math. 46, 299-310, 2022.
- [13] T. M. Baran and M. Kula Local pre-Hausdorff extended pseudo-quasi-semi metric spaces. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68 (1), 862-870, 2019.
- [14] R. Beattie, and H. P. Butzmann, Convergence Structers and Applications to Functional Analysis, Kluwer Academic Publishers, 2002.
- [15] M. M. Clementino, E. Giuli and W. Tholen, Topology in a category: Compactness, Portugal Math. 53, 129-143, 1996.
- [16] H. Herrlich, G. Salicrup and G. E. Strecker, Factorizations, denseness, separation, and relatively compact objects, Topology Appl. 27, 157-169, 1987.
- [17] M. Kula, A note on Cauchy spaces, Acta Math. Hungar. 133, 14-32, 2011.
- [18] M. Kula and T. M. Baran, Separation axioms, Urysohn’s Lemma and Tietze Extention Theorem for extended pseudo-quasi-semi metric spaces, Filomat, 36 (2), 703-713, 2022.
- [19] S. Kula and M. Kula, Seperation, irreducibility, Urysohn’s lemma and Tietze extension theorem for Cauchy spaces, Filomat, 37, 6417-6426, 2023.
- [20] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps, M. Dekker, New York, 1989.
- [21] E. Lowen-Colebunders On composition closed function classes, Acta Math. Acad. Sci. Hungar. 44, 181-189, 1984.
- [22] E. G. Manes, Compact Hausdorff objects, Gen. Topology Appl. 4, 341-360, 1974.
- [23] M. V. Mielke, Convenient categories for internal singular algebraic topology, Illinois J. Math. 27, 519-534, 1983.
- [24] M. V. Mielke, The interval in algebraic topology, Illinois J. Math. 25, 51-62, 1981.
- [25] M. V. Mielke, Separation, axioms and geometric realizations, Indian J. Pure Appl. Math. 27 (7), 711-722, 1994.
- [26] G. Preuss, Theory of Topological Structures, An Approach to topological Categories, D. Reidel Publ. Co., Dordrecht, 1988.
- [27] J. Stine, Pre-Hausdorff objects in topological categories, PhD Dissertation, University of Miami, 1997.
Year 2025,
, 928 - 938, 24.06.2025
Abstract
References
- [1] J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.
- [2] M. Baran, Separation properties, Indian J. Pure Appl. Math. 23 (5), 333-341, 1991.
- [3] M. Baran, Stacks and filters, Turk. J. Math. 16, 95-108, 1992.
- [4] M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hungar. 80, 211-224, 1998.
- [5] M. Baran, Pre$T_2$ Objects in topological categories, Appl. Categor. Struct. 17, 591-602, 2009.
- [6] T. M. Baran, Closedness, separation and connectedness in pseudo-quasi-semi metric spaces, Filomat, 34 (14), 4757-4766, 2020.
- [7] M. Baran, Separation, connectedness and disconnectedness, Turk. J. Math. 47, 279- 295, 2023.
- [8] M. Baran, Stone spaces I, Filomat, 38 (16), 2024.
- [9] M. Baran and H. Altindis, $T_0$-objects in topological categories, J. Univ. Kuwait 22, 123-127, 1995.
- [10] M. Baran and H. Altindis, $T_2$-objects in topological categories, Acta Math. Hungar. 71, 41-48, 1996.
- [11] M. Baran and J. Al-Safar, Quotient-reflective and bireflective subcategories of the category of preordered sets, Topology Appl. 158, 2076-2084, 2011.
- [12] M. Baran and H. Ebughalwa, Sober spaces, Turk. J. Math. 46, 299-310, 2022.
- [13] T. M. Baran and M. Kula Local pre-Hausdorff extended pseudo-quasi-semi metric spaces. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68 (1), 862-870, 2019.
- [14] R. Beattie, and H. P. Butzmann, Convergence Structers and Applications to Functional Analysis, Kluwer Academic Publishers, 2002.
- [15] M. M. Clementino, E. Giuli and W. Tholen, Topology in a category: Compactness, Portugal Math. 53, 129-143, 1996.
- [16] H. Herrlich, G. Salicrup and G. E. Strecker, Factorizations, denseness, separation, and relatively compact objects, Topology Appl. 27, 157-169, 1987.
- [17] M. Kula, A note on Cauchy spaces, Acta Math. Hungar. 133, 14-32, 2011.
- [18] M. Kula and T. M. Baran, Separation axioms, Urysohn’s Lemma and Tietze Extention Theorem for extended pseudo-quasi-semi metric spaces, Filomat, 36 (2), 703-713, 2022.
- [19] S. Kula and M. Kula, Seperation, irreducibility, Urysohn’s lemma and Tietze extension theorem for Cauchy spaces, Filomat, 37, 6417-6426, 2023.
- [20] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps, M. Dekker, New York, 1989.
- [21] E. Lowen-Colebunders On composition closed function classes, Acta Math. Acad. Sci. Hungar. 44, 181-189, 1984.
- [22] E. G. Manes, Compact Hausdorff objects, Gen. Topology Appl. 4, 341-360, 1974.
- [23] M. V. Mielke, Convenient categories for internal singular algebraic topology, Illinois J. Math. 27, 519-534, 1983.
- [24] M. V. Mielke, The interval in algebraic topology, Illinois J. Math. 25, 51-62, 1981.
- [25] M. V. Mielke, Separation, axioms and geometric realizations, Indian J. Pure Appl. Math. 27 (7), 711-722, 1994.
- [26] G. Preuss, Theory of Topological Structures, An Approach to topological Categories, D. Reidel Publ. Co., Dordrecht, 1988.
- [27] J. Stine, Pre-Hausdorff objects in topological categories, PhD Dissertation, University of Miami, 1997.
There are 27 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Topology |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | January 27, 2025 |
| Publication Date | June 24, 2025 |
| Submission Date | February 24, 2024 |
| Acceptance Date | August 20, 2024 |
| Published in Issue | Year 2025 |