About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Kula, Muammer ; Özkan, Samed
$T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces. (English). Mathematica Bohemica, vol. 145 (2020), issue 2, pp. 177-190
MSC: 18B99, 54B30, 54D10, 54E05 | MR 4221828 | Zbl 07217188 | DOI: 10.21136/MB.2019.0144-17
Keywords:
topological category; proximity space; Hausdorff space; regular space
Summary:
In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point $p$ in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various ${\rm Pre}T_{2}$, $T_{i}$, $i=0,1,2,3$, structures at a point $p$ are investigated. Finally, we examine the relationships between the generalized separation properties and the separation properties at a point $p$ in this category.
References:
[1] Adámek, J., Herrlich, H., Strecker, G. E.: Abstract and Concrete Categories: The Joy of Cats. Repr. Theory Appl. Categ. No. 17 {\it 2006} (2006), 1-507. MR 2240597 | Zbl 1113.18001
[2] Baran, M.: Separation properties. Indian J. Pure Appl. Math. 23 (1992), 333-341. MR 1166899 | Zbl 0767.54014
[3] Baran, M.: The notion of closedness in topological categories. Commentat. Math. Univ. Carolin. 34 (1993), 383-395. MR 1241748 | Zbl 0780.18003
[4] Baran, M.: Generalized local separation properties. Indian J. Pure Appl. Math. 25 (1994), 615-620. MR 1285223 | Zbl 0830.18001
[5] Baran, M.: Separation properties in topological categories. Math. Balk., New Ser. 10 (1996), 39-48. MR 1429148 | Zbl 1036.54502
[6] Baran, M.: A notion of compactness in topological categories. Publ. Math. 50 (1997), 221-234. MR 1446467 | Zbl 0880.54010
[7] Baran, M.: $T_3$ and $T_4$-objects in topological categories. Indian J. Pure Appl. Math. 29 (1998), 59-69. MR 1613340 | Zbl 0920.54009
[8] Baran, M.: Closure operators in convergence spaces. Acta Math. Hung. 87 (2000), 33-45. DOI 10.1023/A:1006768916033 | MR 1755877 | Zbl 0963.54003
[9] Baran, M.: $ PreT_2$ objects in topological categories. Appl. Categ. Struct. 17 (2009), 591-602. DOI 10.1007/s10485-008-9161-4 | MR 2564124 | Zbl 1196.54018
[10] Baran, M., Al-Safar, J.: Quotient-reflective and bireflective subcategories of the category of preordered sets. Topology Appl. 158 (2011), 2076-2084. DOI 10.1016/j.topol.2011.06.043 | MR 2825362 | Zbl 1226.54015
[11] Baran, M., Altindis, H.: $T_2$-objects in topological categories. Acta Math. Hung. 71 (1996), 41-48. DOI 10.1007/BF00052193 | MR 1398023 | Zbl 0857.18002
[12] Baran, M., Kula, M.: A note on connectedness. Publ. Math. 68 (2006), 489-501. MR 2212333 | Zbl 1099.54019
[13] Baran, M., Kula, S., Baran, T. M., Qasim, M.: Closure operators in semiuniform convergence spaces. Filomat 30 (2016), 131-140. DOI 10.2298/FIL1601131B | MR 3498758 | Zbl 06749669
[14] Dikranjan, D., Giuli, E.: Closure operators I. Topology Appl. 27 (1987), 129-143. DOI 10.1016/0166-8641(87)90100-3 | MR 0911687 | Zbl 0634.54008
[15] Efremovich, V. A.: Infinitesimal spaces. Dokl. Akad. Nauk SSSR, N. Ser. 76 (1951), 341-343 Russian. MR 0040748 | Zbl 0042.16703
[16] Efremovich, V. A.: The geometry of proximity. I. Mat. Sb., N. Ser. 31(73) (1952), 189-200 Russian. MR 0055659 | Zbl 0046.16302
[17] Friedler, L.: Quotients of proximity spaces. Proc. Am. Math. Soc. 37 (1973), 589-594. DOI 10.2307/2039491 | MR 0402691 | Zbl 0228.54021
[18] Hunsaker, W. N., Sharma, P. L.: Proximity spaces and topological functors. Proc. Am. Math. Soc. 45 (1974), 419-425. DOI 10.2307/2039971 | MR 0353265 | Zbl 0261.54019
[19] Johnstone, P. T.: Topos Theory. London Mathematical Society Monographs, Vol. 10. Academic Press, London (1977). MR 0470019 | Zbl 0368.18001
[20] Kula, M.: Separation properties at $p$ for the topological category of Cauchy spaces. Acta Math. Hung. 136 (2012), 1-15. DOI 10.1007/s10474-012-0238-z | MR 2925752 | Zbl 1274.54048
[21] Kula, M., Maraşlı, T., Özkan, S.: A note on closedness and connectedness in the category of proximity spaces. Filomat 28 (2014), 1483-1492. DOI 10.2298/FIL1407483K | MR 3360054 | Zbl 06704866
[22] Kula, M., Özkan, S., Maraşlı, T.: Pre-Hausdorff and Hausdorff proximity spaces. Filomat 31 (2017), 3837-3846. DOI 10.2298/fil1712837k | MR 3703876
[23] Naimpally, S. A., Warrack, B. D.: Proximity Spaces. Cambridge Tracts in Mathematics and Mathematical Physics, No. 59. Cambridge University Press, London (1970). MR 0278261 | Zbl 0206.24601
[24] Preuss, G.: Theory of Topological Structures. An Approach to Categorical Topology. Mathematics and Its Applications 39. D. Reidel Publishing Company, Dordrecht (1988). DOI 10.1007/978-94-009-2859-6 | MR 0937052 | Zbl 0649.54001
[25] Sharma, P. L.: Proximity bases and subbases. Pac. J. Math. 37 (1971), 515-526. DOI 10.2140/pjm.1971.37.515 | MR 0305358 | Zbl 0216.19301
[26] Smirnov, Y. M.: On proximity spaces. Mat. Sb., N. Ser. 31(73) (1952), 543-574 Russian. MR 0055661 | Zbl 0047.41903
[27] Willard, S.: General Topology. Addison-Wesley Series in Mathematics. Publication Data Reading, Addison-Wesley Publishing Company, Massachusetts (1970). MR 0264581 | Zbl 0205.26601
Partner of
EuDML logo