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Year 2021, Volume: 42 Issue: 2, 339 - 345, 30.06.2021

Jiarul Hoque , Shyamapada Modak

https://doi.org/10.17776/csj.784074

Abstract

References

  • [1] Levine N., Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963) 36-41.
  • [2] Ahmad B., Khan M., Noiri T., A note on semi-frontier, Indian J. Pure Appl. Math., 22 (1) (1991) 61-62.
  • [3] Caldas M., Semi- -spaces, Pro. Math., 8 (1994) 115-121.
  • [4] Caldas M., A separation axiom between semi- and semi- , Mem. Fac. Sci. Kochi Univ. (Math.), 18 (1997) 37-42.
  • [5] Crossley S.G., Hildebrand S.K., Semi-closure, Texas J. Sci., 22 (1971) 99-112.
  • [6] Das P., Note on some applications of semi-open sets, Progr. Math., 7 (1973) 33-44.
  • [7] Davis A.S., Indexed symtems of neighbourhoods for general topological spaces, Amer. Math. Monthly, 68 (1961) 886-894.
  • [8] Maio G.D., On semi topological operators and semi separation axioms, Rend. Circ. Mat. Palermo (2) Suppl. Second Topology Conference (4 Taormina 1984), 12 (1986) 219-230.
  • [9] Jankovic D.D., Reilly I.L., On semi separation properties, Indian J. Pure Appl. Math., 16 (9) (1985) 957-964.
  • [10] Maheshwari S.N., Prasad R., Some new separation axioms, Ann. Soc. Sci. Bruxelles, 89 (1975) 395-402.
  • [11] Sundaram P., Maki H., Balachandran K., Semi-generalized continuous maps and semi- spaces, Bull. Fukuoka Univ. Ed. Part III, 40 (1991) 33-40.
  • [12] Maki H., Generalized -sets and the associated closure operator, The Special Issue in Commemoration of Prof. Kazusada Ikeda’s Retirement, (1986) 139-146.
  • [13] Dontvhev J., Maki H., On sg-closed sets and semi- -closed sets, Questions Answers Gen. Topology, 15 (2) (1997) 259-266.
  • [14] Maheshwari S.N., Prasad R., On -spaces, Port. Math., 34 (1975) 213-217.
  • [15] Bhattacharya P., Lahiri B.K., Semi-generalized closed set in topology, Indian J. Math., 29 (1987) 375-382.
  • [16] Gabai H., The exterior operator and boundary operator, Amer. Math. Monthly, 71 (9) (1964) 1029-1031.
  • [17] Khodabocus M.I., Sookia N.U.H., Theory of generalized exterior and generalized frontier operators in generalized topological spaces: Definitions, Essential Properties and, Consistent, Independent Axioms, Research Series in Pure Mathematics Topology. Exterior and Frontier Operators, Series., 7 (2018-2019) 1-59.
  • [18] Kleiner I.Z., Closure and boundary operators in topological spaces, Ukr. Math. J., 29 (1977) 295-296.
  • [19] Modak S., Some points on generalized open sets, Casp. J. Math. Sci., 6 (2) (2017) 99-106.
  • [20] Modak S., Hoque J., Sk Selim., Homeomorphic image of some kernels, Cankaya Uni. J. Sci. Eng., 17 (1) (2020) 052-062.
  • [21] Nour T.M., A note on some applications of semi-open sets, Int. J. Math. & Math. Sci., 21 (1998) 205-207.
  • [22] Sk Selim., Modak S., Islam Md.M., Characterizations of Hayashi-Samuel spaces via boundary points, Commun. Adv. Math. Sci., II (3) (2019) 219-226.
  • [23] Cueva M.C., Dontchev J., G. -sets and G. -sets, arXiv:math/9810080v1, (1998).
  • [24] Sabah A., Khan M., Kočinac L. D. R., Covering properties defined by semi-open sets, J. Nonlinear Sci. Appl., 9 (2016) 4388-4398.

Algebra of frontier points via semi-kernels

Year 2021, Volume: 42 Issue: 2, 339 - 345, 30.06.2021

Jiarul Hoque , Shyamapada Modak

https://doi.org/10.17776/csj.784074

Abstract

In topological spaces, the study of interior and closure of a set are renowned concepts where the interior is defined as the union of open sets and the closure is defined as the intersection of closed sets. In literature, it is also a significant study while a set is defined as the intersection of open sets, and the union of closed sets. These respective ideas are known as the kernel of a set and its complementary function. Utilizing these ideas, some authors have introduced various kinds of results in topological spaces. Some mathematicians have extended these concepts via Levine’s semi-open sets to semi-kernel and its complementary function. The study of these notions is also a remarkable part of the field of topological spaces as the collection of semi-open sets does not form a topology again. In this paper, we have taken the semi-kernel and its complementary function into account to introduce new types of frontier points. After that we have studied and presented several characterizations of these new types of frontiers and established relationships among them. Finally, we have shown that semi-homeomorphic images of these new types of frontiers are invariant.

Keywords

$\bigvee_s$-set $\bigwedge_s$-set Frontier point Semi-homeomorphism

References

  • [1] Levine N., Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963) 36-41.
  • [2] Ahmad B., Khan M., Noiri T., A note on semi-frontier, Indian J. Pure Appl. Math., 22 (1) (1991) 61-62.
  • [3] Caldas M., Semi- -spaces, Pro. Math., 8 (1994) 115-121.
  • [4] Caldas M., A separation axiom between semi- and semi- , Mem. Fac. Sci. Kochi Univ. (Math.), 18 (1997) 37-42.
  • [5] Crossley S.G., Hildebrand S.K., Semi-closure, Texas J. Sci., 22 (1971) 99-112.
  • [6] Das P., Note on some applications of semi-open sets, Progr. Math., 7 (1973) 33-44.
  • [7] Davis A.S., Indexed symtems of neighbourhoods for general topological spaces, Amer. Math. Monthly, 68 (1961) 886-894.
  • [8] Maio G.D., On semi topological operators and semi separation axioms, Rend. Circ. Mat. Palermo (2) Suppl. Second Topology Conference (4 Taormina 1984), 12 (1986) 219-230.
  • [9] Jankovic D.D., Reilly I.L., On semi separation properties, Indian J. Pure Appl. Math., 16 (9) (1985) 957-964.
  • [10] Maheshwari S.N., Prasad R., Some new separation axioms, Ann. Soc. Sci. Bruxelles, 89 (1975) 395-402.
  • [11] Sundaram P., Maki H., Balachandran K., Semi-generalized continuous maps and semi- spaces, Bull. Fukuoka Univ. Ed. Part III, 40 (1991) 33-40.
  • [12] Maki H., Generalized -sets and the associated closure operator, The Special Issue in Commemoration of Prof. Kazusada Ikeda’s Retirement, (1986) 139-146.
  • [13] Dontvhev J., Maki H., On sg-closed sets and semi- -closed sets, Questions Answers Gen. Topology, 15 (2) (1997) 259-266.
  • [14] Maheshwari S.N., Prasad R., On -spaces, Port. Math., 34 (1975) 213-217.
  • [15] Bhattacharya P., Lahiri B.K., Semi-generalized closed set in topology, Indian J. Math., 29 (1987) 375-382.
  • [16] Gabai H., The exterior operator and boundary operator, Amer. Math. Monthly, 71 (9) (1964) 1029-1031.
  • [17] Khodabocus M.I., Sookia N.U.H., Theory of generalized exterior and generalized frontier operators in generalized topological spaces: Definitions, Essential Properties and, Consistent, Independent Axioms, Research Series in Pure Mathematics Topology. Exterior and Frontier Operators, Series., 7 (2018-2019) 1-59.
  • [18] Kleiner I.Z., Closure and boundary operators in topological spaces, Ukr. Math. J., 29 (1977) 295-296.
  • [19] Modak S., Some points on generalized open sets, Casp. J. Math. Sci., 6 (2) (2017) 99-106.
  • [20] Modak S., Hoque J., Sk Selim., Homeomorphic image of some kernels, Cankaya Uni. J. Sci. Eng., 17 (1) (2020) 052-062.
  • [21] Nour T.M., A note on some applications of semi-open sets, Int. J. Math. & Math. Sci., 21 (1998) 205-207.
  • [22] Sk Selim., Modak S., Islam Md.M., Characterizations of Hayashi-Samuel spaces via boundary points, Commun. Adv. Math. Sci., II (3) (2019) 219-226.
  • [23] Cueva M.C., Dontchev J., G. -sets and G. -sets, arXiv:math/9810080v1, (1998).
  • [24] Sabah A., Khan M., Kočinac L. D. R., Covering properties defined by semi-open sets, J. Nonlinear Sci. Appl., 9 (2016) 4388-4398.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Jiarul Hoque 0000-0003-1055-9820

Shyamapada Modak 0000-0002-0226-2392

Publication Date June 30, 2021
Submission Date August 22, 2020
Acceptance Date May 31, 2021
Published in Issue Year 2021Volume: 42 Issue: 2

Cite

APA Hoque, J., & Modak, S. (2021). Algebra of frontier points via semi-kernels. Cumhuriyet Science Journal, 42(2), 339-345. https://doi.org/10.17776/csj.784074