Geodesic
$T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces
Mathematica Bohemica, Tome 145 (2020) no. 2, pp. 177-190
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/MB.2019.0144-17
Classification : 18B99, 54B30, 54D10, 54E05
Keywords: topological category; proximity space; Hausdorff space; regular space 
@article{10_21136_MB_2019_0144_17,
     author = {Kula, Muammer and \"Ozkan, Samed},
     title = {$T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces},
     journal = {Mathematica Bohemica},
     pages = {177--190},
     year = {2020},
     volume = {145},
     number = {2},
     doi = {10.21136/MB.2019.0144-17},
     mrnumber = {4221828},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0144-17/}
}
TY  - JOUR
AU  - Kula, Muammer
AU  - Özkan, Samed
TI  - $T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces
JO  - Mathematica Bohemica
PY  - 2020
SP  - 177
EP  - 190
VL  - 145
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0144-17/
DO  - 10.21136/MB.2019.0144-17
LA  - en
ID  - 10_21136_MB_2019_0144_17
ER  - 
%0 Journal Article
%A Kula, Muammer
%A Özkan, Samed
%T $T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces
%J Mathematica Bohemica
%D 2020
%P 177-190
%V 145
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2019.0144-17/
%R 10.21136/MB.2019.0144-17
%G en
%F 10_21136_MB_2019_0144_17
Kula, Muammer; Özkan, Samed. $T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces. Mathematica Bohemica, Tome 145 (2020) no. 2, pp. 177-190. doi: 10.21136/MB.2019.0144-17

[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 | JFM

[2] Baran, M.: Separation properties. Indian J. Pure Appl. Math. 23 (1992), 333-341. | MR | JFM

[3] Baran, M.: The notion of closedness in topological categories. Commentat. Math. Univ. Carolin. 34 (1993), 383-395. | MR | JFM

[4] Baran, M.: Generalized local separation properties. Indian J. Pure Appl. Math. 25 (1994), 615-620. | MR | JFM

[5] Baran, M.: Separation properties in topological categories. Math. Balk., New Ser. 10 (1996), 39-48. | MR | JFM

[6] Baran, M.: A notion of compactness in topological categories. Publ. Math. 50 (1997), 221-234. | MR | JFM

[7] Baran, M.: $T_3$ and $T_4$-objects in topological categories. Indian J. Pure Appl. Math. 29 (1998), 59-69. | MR | JFM

[8] Baran, M.: Closure operators in convergence spaces. Acta Math. Hung. 87 (2000), 33-45. | DOI | MR | JFM

[9] Baran, M.: $ PreT_2$ objects in topological categories. Appl. Categ. Struct. 17 (2009), 591-602. | DOI | MR | JFM

[10] Baran, M., Al-Safar, J.: Quotient-reflective and bireflective subcategories of the category of preordered sets. Topology Appl. 158 (2011), 2076-2084. | DOI | MR | JFM

[11] Baran, M., Altindis, H.: $T_2$-objects in topological categories. Acta Math. Hung. 71 (1996), 41-48. | DOI | MR | JFM

[12] Baran, M., Kula, M.: A note on connectedness. Publ. Math. 68 (2006), 489-501. | MR | JFM

[13] Baran, M., Kula, S., Baran, T. M., Qasim, M.: Closure operators in semiuniform convergence spaces. Filomat 30 (2016), 131-140. | DOI | MR | JFM

[14] Dikranjan, D., Giuli, E.: Closure operators I. Topology Appl. 27 (1987), 129-143. | DOI | MR | JFM

[15] Efremovich, V. A.: Infinitesimal spaces. Dokl. Akad. Nauk SSSR, N. Ser. 76 (1951), 341-343 Russian. | MR | JFM

[16] Efremovich, V. A.: The geometry of proximity. I. Mat. Sb., N. Ser. 31(73) (1952), 189-200 Russian. | MR | JFM

[17] Friedler, L.: Quotients of proximity spaces. Proc. Am. Math. Soc. 37 (1973), 589-594. | DOI | MR | JFM

[18] Hunsaker, W. N., Sharma, P. L.: Proximity spaces and topological functors. Proc. Am. Math. Soc. 45 (1974), 419-425. | DOI | MR | JFM

[19] Johnstone, P. T.: Topos Theory. London Mathematical Society Monographs, Vol. 10. Academic Press, London (1977). | MR | JFM

[20] Kula, M.: Separation properties at $p$ for the topological category of Cauchy spaces. Acta Math. Hung. 136 (2012), 1-15. | DOI | MR | JFM

[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 | MR | JFM

[22] Kula, M., Özkan, S., Maraşlı, T.: Pre-Hausdorff and Hausdorff proximity spaces. Filomat 31 (2017), 3837-3846. | DOI | MR

[23] Naimpally, S. A., Warrack, B. D.: Proximity Spaces. Cambridge Tracts in Mathematics and Mathematical Physics, No. 59. Cambridge University Press, London (1970). | MR | JFM

[24] Preuss, G.: Theory of Topological Structures. An Approach to Categorical Topology. Mathematics and Its Applications 39. D. Reidel Publishing Company, Dordrecht (1988). | DOI | MR | JFM

[25] Sharma, P. L.: Proximity bases and subbases. Pac. J. Math. 37 (1971), 515-526. | DOI | MR | JFM

[26] Smirnov, Y. M.: On proximity spaces. Mat. Sb., N. Ser. 31(73) (1952), 543-574 Russian. | MR | JFM

[27] Willard, S.: General Topology. Addison-Wesley Series in Mathematics. Publication Data Reading, Addison-Wesley Publishing Company, Massachusetts (1970). | MR | JFM

Cité par Sources :