T 2 and T 3 objects at p in the category of proximity spaces

Muammer Kula; Samed Özkan

Mathematica Bohemica (2020)

  • Volume: 145, Issue: 2, page 177-190
  • ISSN: 0862-7959

Abstract

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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 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.

How to cite

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Kula, Muammer, and Özkan, Samed. "$T_{2}$ and $T_{3}$ objects at $p$ in the category of proximity spaces." Mathematica Bohemica 145.2 (2020): 177-190. <http://eudml.org/doc/297397>.

@article{Kula2020,
abstract = {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.},
author = {Kula, Muammer, Özkan, Samed},
journal = {Mathematica Bohemica},
keywords = {topological category; proximity space; Hausdorff space; regular space},
language = {eng},
number = {2},
pages = {177-190},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$T_\{2\}$ and $T_\{3\}$ objects at $p$ in the category of proximity spaces},
url = {http://eudml.org/doc/297397},
volume = {145},
year = {2020},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 2
SP - 177
EP - 190
AB - 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.
LA - eng
KW - topological category; proximity space; Hausdorff space; regular space
UR - http://eudml.org/doc/297397
ER -

References

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  1. Adámek, J., Herrlich, H., Strecker, G. E., Abstract and Concrete Categories: The Joy of Cats, Repr. Theory Appl. Categ. No. 17 (2006), 1-507. (2006) Zbl1113.18001MR2240597
  2. Baran, M., Separation properties, Indian J. Pure Appl. Math. 23 (1992), 333-341. (1992) Zbl0767.54014MR1166899
  3. Baran, M., The notion of closedness in topological categories, Commentat. Math. Univ. Carolin. 34 (1993), 383-395. (1993) Zbl0780.18003MR1241748
  4. Baran, M., Generalized local separation properties, Indian J. Pure Appl. Math. 25 (1994), 615-620. (1994) Zbl0830.18001MR1285223
  5. Baran, M., Separation properties in topological categories, Math. Balk., New Ser. 10 (1996), 39-48. (1996) Zbl1036.54502MR1429148
  6. Baran, M., A notion of compactness in topological categories, Publ. Math. 50 (1997), 221-234. (1997) Zbl0880.54010MR1446467
  7. Baran, M., T 3 and T 4 -objects in topological categories, Indian J. Pure Appl. Math. 29 (1998), 59-69. (1998) Zbl0920.54009MR1613340
  8. Baran, M., 10.1023/A:1006768916033, Acta Math. Hung. 87 (2000), 33-45. (2000) Zbl0963.54003MR1755877DOI10.1023/A:1006768916033
  9. Baran, M., 10.1007/s10485-008-9161-4, Appl. Categ. Struct. 17 (2009), 591-602. (2009) Zbl1196.54018MR2564124DOI10.1007/s10485-008-9161-4
  10. Baran, M., Al-Safar, J., 10.1016/j.topol.2011.06.043, Topology Appl. 158 (2011), 2076-2084. (2011) Zbl1226.54015MR2825362DOI10.1016/j.topol.2011.06.043
  11. Baran, M., Altindis, H., 10.1007/BF00052193, Acta Math. Hung. 71 (1996), 41-48. (1996) Zbl0857.18002MR1398023DOI10.1007/BF00052193
  12. Baran, M., Kula, M., A note on connectedness, Publ. Math. 68 (2006), 489-501. (2006) Zbl1099.54019MR2212333
  13. Baran, M., Kula, S., Baran, T. M., Qasim, M., 10.2298/FIL1601131B, Filomat 30 (2016), 131-140. (2016) Zbl06749669MR3498758DOI10.2298/FIL1601131B
  14. Dikranjan, D., Giuli, E., 10.1016/0166-8641(87)90100-3, Topology Appl. 27 (1987), 129-143. (1987) Zbl0634.54008MR0911687DOI10.1016/0166-8641(87)90100-3
  15. Efremovich, V. A., Infinitesimal spaces, Dokl. Akad. Nauk SSSR, N. Ser. 76 (1951), 341-343 Russian. (1951) Zbl0042.16703MR0040748
  16. Efremovich, V. A., The geometry of proximity. I, Mat. Sb., N. Ser. 31(73) (1952), 189-200 Russian. (1952) Zbl0046.16302MR0055659
  17. Friedler, L., 10.2307/2039491, Proc. Am. Math. Soc. 37 (1973), 589-594. (1973) Zbl0228.54021MR0402691DOI10.2307/2039491
  18. Hunsaker, W. N., Sharma, P. L., 10.2307/2039971, Proc. Am. Math. Soc. 45 (1974), 419-425. (1974) Zbl0261.54019MR0353265DOI10.2307/2039971
  19. Johnstone, P. T., Topos Theory, London Mathematical Society Monographs, Vol. 10. Academic Press, London (1977). (1977) Zbl0368.18001MR0470019
  20. Kula, M., 10.1007/s10474-012-0238-z, Acta Math. Hung. 136 (2012), 1-15. (2012) Zbl1274.54048MR2925752DOI10.1007/s10474-012-0238-z
  21. Kula, M., Maraşlı, T., Özkan, S., 10.2298/FIL1407483K, Filomat 28 (2014), 1483-1492. (2014) Zbl06704866MR3360054DOI10.2298/FIL1407483K
  22. Kula, M., Özkan, S., Maraşlı, T., 10.2298/fil1712837k, Filomat 31 (2017), 3837-3846. (2017) MR3703876DOI10.2298/fil1712837k
  23. Naimpally, S. A., Warrack, B. D., Proximity Spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59. Cambridge University Press, London (1970). (1970) Zbl0206.24601MR0278261
  24. Preuss, G., 10.1007/978-94-009-2859-6, Mathematics and Its Applications 39. D. Reidel Publishing Company, Dordrecht (1988). (1988) Zbl0649.54001MR0937052DOI10.1007/978-94-009-2859-6
  25. Sharma, P. L., 10.2140/pjm.1971.37.515, Pac. J. Math. 37 (1971), 515-526. (1971) Zbl0216.19301MR0305358DOI10.2140/pjm.1971.37.515
  26. Smirnov, Y. M., On proximity spaces, Mat. Sb., N. Ser. 31(73) (1952), 543-574 Russian. (1952) Zbl0047.41903MR0055661
  27. Willard, S., General Topology, Addison-Wesley Series in Mathematics. Publication Data Reading, Addison-Wesley Publishing Company, Massachusetts (1970). (1970) Zbl0205.26601MR0264581

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