Previously, several notions of $ T_{0} $ and $ T_{1} $ objects have been studied and examined in various topological categories. In this paper, we characterize each of $ T_{0} $ and $ T_{1} $ objects in the categories of several types of bounded uniform filter spaces and examine their mutual relations, and compare that with the usual ones. Moreover, it is shown that under $ T_{0} $ (resp. $ T_{1} $) condition, the category of preuniform (resp. semiuniform) convergence spaces and the category of bornological (resp. symmetric) bounded uniform filter spaces are isomorphic. Finally, it is proved that the category of each of $ T_{0} $ (resp. $ T_{1} $) bounded uniform filter space are quotient reflective subcategories of the category of bounded uniform filter spaces.
Citation: Sana Khadim, Muhammad Qasim. Quotient reflective subcategories of the category of bounded uniform filter spaces[J]. AIMS Mathematics, 2022, 7(9): 16632-16648. doi: 10.3934/math.2022911
Previously, several notions of $ T_{0} $ and $ T_{1} $ objects have been studied and examined in various topological categories. In this paper, we characterize each of $ T_{0} $ and $ T_{1} $ objects in the categories of several types of bounded uniform filter spaces and examine their mutual relations, and compare that with the usual ones. Moreover, it is shown that under $ T_{0} $ (resp. $ T_{1} $) condition, the category of preuniform (resp. semiuniform) convergence spaces and the category of bornological (resp. symmetric) bounded uniform filter spaces are isomorphic. Finally, it is proved that the category of each of $ T_{0} $ (resp. $ T_{1} $) bounded uniform filter space are quotient reflective subcategories of the category of bounded uniform filter spaces.
| [1] | J. Adámek, H. Herrlich, G. E. Strecker, Abstract and concrete categories, New York: Wiley, 1990. |
| [2] | M. Baran, Separation properties, Indian J. Pure Appl. Math., 23 (1991), 333–341. |
| [3] | M. Baran, Generalized local separation properties, Indian J. Pure Appl. Math., 25 (1994), 615–620. |
| [4] | M. Baran, H. Altındis, $T_{0}$-objects in topological categories, J. Univ. Kuwait (Sci.), 22 (1995), 123–127. |
| [5] | M. Baran, Separation properties in topological categories, Math. Balkanica, 10 (1996), 39–48. |
| [6] |
M. Baran, Completely regular objects and normal objects in topological categories, Acta Math. Hung., 80 (1998), 211–224. https://doi.org/10.1023/A:1006550726143 doi: 10.1023/A:1006550726143
|
| [7] |
M. Baran, Compactness, perfectness, separation, minimality and closedness with respect to closure operators, Appl. Categor. Struct., 10 (2002), 403–415. https://doi.org/10.1023/A:1016388102703 doi: 10.1023/A:1016388102703
|
| [8] |
M. Baran, Pre$T_2$ objects in topological categories, Appl. Categor. Struct., 17 (2009), 591–602. https://doi.org/10.1007/s10485-008-9161-4 doi: 10.1007/s10485-008-9161-4
|
| [9] | M. Baran, S. Kula, A. Erciyes, $T_{0}$ and $T_{1}$ semiuniform convergence spaces, Filomat, 27 (2013), 537–546. |
| [10] | M. Baran, S. Kula, T. M. Baran, M. Qasim, Closure operators in semiuniform convergence spaces, Filomat, 30 (2016), 131–140. |
| [11] |
M. Baran, H. Abughalwa, Sober spaces, Turkish J. Math., 46 (2022), 299–310. https://doi.org/10.3906/mat-2109-95 doi: 10.3906/mat-2109-95
|
| [12] |
T. M. Baran, Closedness, separation and connectedness in pseudo-quasi-semi metric spaces, Filomat, 34 (2020), 4757–4766. https://doi.org/10.2298/FIL2014757B doi: 10.2298/FIL2014757B
|
| [13] |
T. M. Baran, A. Erciyes, $T_{4}$, Urysohn's lemma and Tietze extension theorem for constant filter convergence spaces, Turkish J. Math., 45 (2021), 843–855. https://doi.org/10.3906/mat-2012-101 doi: 10.3906/mat-2012-101
|
| [14] |
T. M. Baran, M. Kula, Separation axioms, Urysohn's lemma and Tietze extension theorem for extended pseudo-quasi-semi metric spaces, Filomat, 36 (2022), 703–713. https://doi.org/10.2298/FIL2202703B doi: 10.2298/FIL2202703B
|
| [15] | G. C. L. Brümmer, A categorial study of initiality in uniform topology, University of Cape Town, 1971. |
| [16] |
D. Dikranjan, E. Giuli, Closure operators Ⅰ, Topology Appl., 27 (1987), 129–143. https://doi.org/10.1016/0166-8641(87)90100-3 doi: 10.1016/0166-8641(87)90100-3
|
| [17] | D. B. Doitchinov, A unified theory of topological spaces, proximity spaces and uniform spaces, Soviet Math. Dokl., 5 (1964), 595–598. |
| [18] | A. Erciyes, T. M. Baran, M. Qasim, Closure operators in constant filter convergence spaces, Konuralp J. Math., 8 (2020), 185–191. |
| [19] | F. Peter, W. F. Lindgren, Quasi-uniform spaces, 1982. |
| [20] |
J. M. Harvey, $T_0$-separation in topological categories, Quaest. Math., 2 (1977), 177–190. https://doi.org/10.1080/16073606.1977.9632541 doi: 10.1080/16073606.1977.9632541
|
| [21] |
G. T. Herman, On topology as applied to image analysis, Comput. Vis. Graph. Image Process., 52 (1990), 409–415. https://doi.org/10.1016/0734-189X(90)90084-9 doi: 10.1016/0734-189X(90)90084-9
|
| [22] | H. Herrlich, Topological structures, Math. Centre Tracts, 52 (1974), 59–122. |
| [23] | R. E. Hoffmann, $(E, M)$-universally topological functors, Habilitationsscheift, Universität Düsseldorf, 1974. |
| [24] |
G. Janelidze, Light morphisms for generalized $T_{0}$-reflections, Topology Appl., 156 (2009), 2109–2115. https://doi.org/10.1016/j.topol.2009.03.031 doi: 10.1016/j.topol.2009.03.031
|
| [25] | P. T. Johnstone, Stone spaces, New York: Academic Press, 1977. |
| [26] | M. Katĕtov, On continuity structures and spaces of mappings, Comment. Math. Univ. Carolin., 6 (1965), 257–278. |
| [27] | D. C. Kent, Convergence functions and their related topologies, Fund. Math., 54 (1964), 125–133. |
| [28] |
V. A. Kovalevsky, Finite topology as applied to image analysis, Comput. Vis. Graph. Image Process., 46 (1989), 141–161. https://doi.org/10.1016/0734-189X(89)90165-5 doi: 10.1016/0734-189X(89)90165-5
|
| [29] |
V. Kovalevsky, R. Kopperman, Some topology-based image processing algorithms, Ann. New York Acad. Sci., 728 (1994), 174–182. https://doi.org/10.1111/j.1749-6632.1994.tb44143.x doi: 10.1111/j.1749-6632.1994.tb44143.x
|
| [30] | S. Kula, Closedness and $T_{0}$, $T_{1}$ objects in the category of preuniform convergence spaces, Ph. D. thesis, Turkey: Erciyes University, 2014. |
| [31] |
D. Leseberg, Z. Vaziry, The quasitopos of b-uniform filter spaces, Math. Appl., 7 (2018), 155–171. https://doi.org/10.13164/ma.2018.13 doi: 10.13164/ma.2018.13
|
| [32] |
D. Leseberg, Z. Vaziry, On the completeness of non-symmetrical uniform convergence with some links to approach spaces, Math. Appl., 8 (2019), 37–57. https://doi.org/10.13164/ma.2019.04 doi: 10.13164/ma.2019.04
|
| [33] | D. Leseberg, Z. Vaziry, Bounded topology, Lap Lambert Academic Publishing, 2019. |
| [34] | T. Marny, Rechts-Bikategoriestrukturen in topologischen Kategorien: Inaugural-dissertation, 1973. |
| [35] | G. Preuß, Semiuniform convergence spaces, Math. Japn., 41 (1995), 465–491. |
| [36] | G. Preuss, Foundations of topology: An approach to convenient topology, Dordrecht: Springer Science & Business Media, 2002. https://doi.org/10.1007/978-94-010-0489-3 |
| [37] |
M. Qasim, M. Baran, H. Abughalwa, Closure operators in convergence approach spaces, Turkish J. Math., 45 (2021), 139–152. https://doi.org/10.3906/mat-2008-65 doi: 10.3906/mat-2008-65
|
| [38] | A. Salibra, A continuum of theories of lambda calculus without semantics, In: Proceedings 16th Annual IEEE Symposium on Logic in Computer Science, 2001,334–343. https://doi.org/10.1109/LICS.2001.932509 |
| [39] | J. E. Stoy, Denotational semantics: the Scott-Strachey approach to programming language theory, MIT Press, 1981. |
| [40] |
S. Weck-Schwarz, $T_{0}$-objects and separated objects in topological categories, Quaest. Math., 14 (1991), 315–325. https://doi.org/10.1080/16073606.1991.9631649 doi: 10.1080/16073606.1991.9631649
|
Sana Khadim, Muhammad Qasim. Quotient reflective subcategories of the category of bounded uniform filter spaces[J]. AIMS Mathematics, 2022, 7(9): 16632-16648. doi: 10.3934/math.2022911
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