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Year 2020, Volume: 8 Issue: 1, 50 - 56, 15.04.2020

Shyamapada Modak Md. Monirul Islam

Abstract

References

  • [1] A. Al-Omari and T. Noiri, Local closure functions in ideal topological spaces, Novi Sad J. Math., 43(2) (2013), 139-149.
  • [2] D. Andrijev´ic, On b-open sets, Mat. Vesnik, 48 (1996), 59-64.
  • [3] D. Andrijev´ic, Semi-preopen sets, Mat. Vesnik, 38 (1986), 24-32.
  • [4] C. Bandyopadhyay and S. Modak, A new topology via y-operator, Proc. Nat. Acad. Sci. India, 76(A)(IV) (2006), 317-320.
  • [5] C. Chattopadhyay and C. Bandyopadhyay, On structure of d-sets, Bull. Cal. Math. Soc., 83 (1991), 281-290.
  • [6] C. Chattopadhyay and U.K. Roy, d-sets, irresolvable and resolvable spaces, Math. Slovaca, 42(3) (1992), 371-378.
  • [7] G. Choquet, Sur les notions de filtre et grille, Comptes Rendus Acad. Sci. Paris, 224 (1947), 171-173.
  • [8] J. Dontchev, Idealization of Ganster-Reilly decomposition theorems, arXIV:math. Gn/9901017v1 [math.GN], 5 Jan 1999.
  • [9] J. Dontchev, On pre-I-open sets and a decomposition of I-continuity, Banyan Math. J., 2 (1996).
  • [10] J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl., 93 (1999), 1-16.
  • [11] J. Dontchev, On Hausdorff spaces via topological ideals and I-irresolute functions, Ann. NY Acad. Sci., 767(1) (1995), 28-38.
  • [12] M.E.A. El-Monsef, S.N. El-Deeb and R.A. Mahmoud, b-open sets and b-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77-90.
  • [13] E. Ekici, On a-open sets, A-sets and decompositions of continuity and supercontinuity, Annales Univ. Sci. Budapest, 51 (2008), 39-51.
  • [14] T.R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator Y , Bollettino U. M. I., 7 (4-B) (1990), 863-874.
  • [15] H. Hashimoto, On the -topology and its applications, Fund. Math., 91 (1976), 5-10.
  • [16] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96 (2002), 341-349.
  • [17] E. Hayashi, Topologies defined by local properties, Math. Ann., 156 (1964), 205-215.
  • [18] E. Hewitt, A problem of set theoretic topology, Duke Math. J., 10(2)(1943), 309-333.
  • [19] D. Jankovi ´ c and T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310.
  • [20] A. Kandil, S.A. El-Sheikh, M. Abdelhakem, S.A. Hazza, On ideals and grills in topological spaces, South Asian J. Math., 5(6), 2015, 233-238.
  • [21] K. Kuratowski, Topology, Vol. I, New York, Academic Press, (1966).
  • [22] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.
  • [23] A.S. Mashhour, M.E.A. El-Monsef, S.N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. [Egypt], 53 (1982), 47-53.
  • [24] S. Modak, Some new topologies on ideal topological spaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 82(3) (2012), 233-243.
  • [25] S. Modak, Minimal spaces with a mathematical structure, J. Assoc. Arab Univ. Basic Appl. Sci., 22 (2017), 98-101.
  • [26] S. Modak, Decompositions of generalized continuity in grill topological spaces, Thai J. Math., 13(2) (2015), 511-518.
  • [27] S. Modak, Grill-filter space, J. Indian Math. Soc., 80 (2013), 313–320.
  • [28] S. Modak, Topology on grill-filter space and continuity, Bol. Soc. Paran. Mat., 31 (2013), 1–12.
  • [29] S. Modak, Topology on grill m-space, Jordan J. Math. Stat., 6 (2013), 183–195.
  • [30] S. Modak, Operators on grill M-space, Bol. Soc. Paran. Mat., 31 (2013), 101-107.
  • [31] S. Modak and C. Bandyopadhyay, -topology and generalized open sets, Soochow J. Math., 32(2)(2006), 201-210.
  • [32] S. Modak and C. Bandyopadhyay, A note on y-operator, Bull. Malyas. Math. Sci. Soc., 30(1) (2007), 43-48.
  • [33] S. Modak and Md.M. Islam, On  and Y operators in topological spaces with ideals, Trans. A. Razmadze Math. Inst., 172 (2018), 491-497.
  • [34] S. Modak and Md.M. Islam, New operators in ideal topological spaces and their closure spaces, Aksaray J. Sci. Eng., 3(2) (2019), 112-128.
  • [35] Md.M. Islam and S. Modak, Operators associated with the  and Y operators, J. Taibah Univ. Sci., 12(4) (2018), 444-449.
  • [36] S. Modak and T. Noiri, Connectedness of ideal topological spaces, Filomat, 29(4) (2015), 661-665.
  • [37] T. Natkaniec, On I-continuity and I-semicontinuity points, Math. Slovaca, 36(3) (1986), 297-312.
  • [38] R.L. Newcomb, Topologies which are compact modulo an ideal, Ph.D. Dissertation, Univ. of Cal. at Santa Barbara, (1967).
  • [39] O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15(3) (1965), 961-970.
  • [40] B. Roy and M.N. Mukherjee, On a typical topology induced by a grill, Soochow J. Math., 33(4) (2007), 771-786.
  • [41] S. Suriyakala and R. Vembu, Relations between union and intersection of ideals and their corresponding ideal topologies, Novi Sad J. Math., 45(2), 39-46.
  • [42] M.H. Stone, Application of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-481.
  • [43] R. Vaidyanathswamy, The localisation theory in set topology, Proc. India Acad. Sci., 20 (1945), 51-61.

Generalized Open Sets vis-a-vis $\Delta$-Sets

Year 2020, Volume: 8 Issue: 1, 50 - 56, 15.04.2020

Shyamapada Modak Md. Monirul Islam

Abstract

This paper concerns about the splitting of the collections of generalized open sets in topological spaces and their decompositions. Several characterizations of these sets are also discussed in this paper. Further, this paper also introduce a new type of normal space and its characterizations. Several properties of this space are discussed.



Keywords

$\Delta$-operator $\Delta$-sets Hayashi-Samuel space $\Psi-*-$normal

References

  • [1] A. Al-Omari and T. Noiri, Local closure functions in ideal topological spaces, Novi Sad J. Math., 43(2) (2013), 139-149.
  • [2] D. Andrijev´ic, On b-open sets, Mat. Vesnik, 48 (1996), 59-64.
  • [3] D. Andrijev´ic, Semi-preopen sets, Mat. Vesnik, 38 (1986), 24-32.
  • [4] C. Bandyopadhyay and S. Modak, A new topology via y-operator, Proc. Nat. Acad. Sci. India, 76(A)(IV) (2006), 317-320.
  • [5] C. Chattopadhyay and C. Bandyopadhyay, On structure of d-sets, Bull. Cal. Math. Soc., 83 (1991), 281-290.
  • [6] C. Chattopadhyay and U.K. Roy, d-sets, irresolvable and resolvable spaces, Math. Slovaca, 42(3) (1992), 371-378.
  • [7] G. Choquet, Sur les notions de filtre et grille, Comptes Rendus Acad. Sci. Paris, 224 (1947), 171-173.
  • [8] J. Dontchev, Idealization of Ganster-Reilly decomposition theorems, arXIV:math. Gn/9901017v1 [math.GN], 5 Jan 1999.
  • [9] J. Dontchev, On pre-I-open sets and a decomposition of I-continuity, Banyan Math. J., 2 (1996).
  • [10] J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl., 93 (1999), 1-16.
  • [11] J. Dontchev, On Hausdorff spaces via topological ideals and I-irresolute functions, Ann. NY Acad. Sci., 767(1) (1995), 28-38.
  • [12] M.E.A. El-Monsef, S.N. El-Deeb and R.A. Mahmoud, b-open sets and b-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77-90.
  • [13] E. Ekici, On a-open sets, A-sets and decompositions of continuity and supercontinuity, Annales Univ. Sci. Budapest, 51 (2008), 39-51.
  • [14] T.R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator Y , Bollettino U. M. I., 7 (4-B) (1990), 863-874.
  • [15] H. Hashimoto, On the -topology and its applications, Fund. Math., 91 (1976), 5-10.
  • [16] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96 (2002), 341-349.
  • [17] E. Hayashi, Topologies defined by local properties, Math. Ann., 156 (1964), 205-215.
  • [18] E. Hewitt, A problem of set theoretic topology, Duke Math. J., 10(2)(1943), 309-333.
  • [19] D. Jankovi ´ c and T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310.
  • [20] A. Kandil, S.A. El-Sheikh, M. Abdelhakem, S.A. Hazza, On ideals and grills in topological spaces, South Asian J. Math., 5(6), 2015, 233-238.
  • [21] K. Kuratowski, Topology, Vol. I, New York, Academic Press, (1966).
  • [22] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.
  • [23] A.S. Mashhour, M.E.A. El-Monsef, S.N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. [Egypt], 53 (1982), 47-53.
  • [24] S. Modak, Some new topologies on ideal topological spaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 82(3) (2012), 233-243.
  • [25] S. Modak, Minimal spaces with a mathematical structure, J. Assoc. Arab Univ. Basic Appl. Sci., 22 (2017), 98-101.
  • [26] S. Modak, Decompositions of generalized continuity in grill topological spaces, Thai J. Math., 13(2) (2015), 511-518.
  • [27] S. Modak, Grill-filter space, J. Indian Math. Soc., 80 (2013), 313–320.
  • [28] S. Modak, Topology on grill-filter space and continuity, Bol. Soc. Paran. Mat., 31 (2013), 1–12.
  • [29] S. Modak, Topology on grill m-space, Jordan J. Math. Stat., 6 (2013), 183–195.
  • [30] S. Modak, Operators on grill M-space, Bol. Soc. Paran. Mat., 31 (2013), 101-107.
  • [31] S. Modak and C. Bandyopadhyay, -topology and generalized open sets, Soochow J. Math., 32(2)(2006), 201-210.
  • [32] S. Modak and C. Bandyopadhyay, A note on y-operator, Bull. Malyas. Math. Sci. Soc., 30(1) (2007), 43-48.
  • [33] S. Modak and Md.M. Islam, On  and Y operators in topological spaces with ideals, Trans. A. Razmadze Math. Inst., 172 (2018), 491-497.
  • [34] S. Modak and Md.M. Islam, New operators in ideal topological spaces and their closure spaces, Aksaray J. Sci. Eng., 3(2) (2019), 112-128.
  • [35] Md.M. Islam and S. Modak, Operators associated with the  and Y operators, J. Taibah Univ. Sci., 12(4) (2018), 444-449.
  • [36] S. Modak and T. Noiri, Connectedness of ideal topological spaces, Filomat, 29(4) (2015), 661-665.
  • [37] T. Natkaniec, On I-continuity and I-semicontinuity points, Math. Slovaca, 36(3) (1986), 297-312.
  • [38] R.L. Newcomb, Topologies which are compact modulo an ideal, Ph.D. Dissertation, Univ. of Cal. at Santa Barbara, (1967).
  • [39] O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15(3) (1965), 961-970.
  • [40] B. Roy and M.N. Mukherjee, On a typical topology induced by a grill, Soochow J. Math., 33(4) (2007), 771-786.
  • [41] S. Suriyakala and R. Vembu, Relations between union and intersection of ideals and their corresponding ideal topologies, Novi Sad J. Math., 45(2), 39-46.
  • [42] M.H. Stone, Application of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-481.
  • [43] R. Vaidyanathswamy, The localisation theory in set topology, Proc. India Acad. Sci., 20 (1945), 51-61.
There are 43 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Shyamapada Modak 0000-0002-0226-2392

Md. Monirul Islam 0000-0003-4748-4690

Publication Date April 15, 2020
Submission Date February 27, 2019
Acceptance Date March 30, 2020
Published in Issue Year 2020 Volume: 8 Issue: 1

Cite

APA Modak, S., & Islam, M. M. (2020). Generalized Open Sets vis-a-vis $\Delta$-Sets. Konuralp Journal of Mathematics, 8(1), 50-56.
AMA Modak S, Islam MM. Generalized Open Sets vis-a-vis $\Delta$-Sets. Konuralp J. Math. April 2020;8(1):50-56.
Chicago Modak, Shyamapada, and Md. Monirul Islam. “Generalized Open Sets Vis-a-Vis $\Delta$-Sets”. Konuralp Journal of Mathematics 8, no. 1 (April 2020): 50-56.
EndNote Modak S, Islam MM (April 1, 2020) Generalized Open Sets vis-a-vis $\Delta$-Sets. Konuralp Journal of Mathematics 8 1 50–56.
IEEE S. Modak and M. M. Islam, “Generalized Open Sets vis-a-vis $\Delta$-Sets”, Konuralp J. Math., vol. 8, no. 1, pp. 50–56, 2020.
ISNAD Modak, Shyamapada - Islam, Md. Monirul. “Generalized Open Sets Vis-a-Vis $\Delta$-Sets”. Konuralp Journal of Mathematics 8/1 (April 2020), 50-56.
JAMA Modak S, Islam MM. Generalized Open Sets vis-a-vis $\Delta$-Sets. Konuralp J. Math. 2020;8:50–56.
MLA Modak, Shyamapada and Md. Monirul Islam. “Generalized Open Sets Vis-a-Vis $\Delta$-Sets”. Konuralp Journal of Mathematics, vol. 8, no. 1, 2020, pp. 50-56.
Vancouver Modak S, Islam MM. Generalized Open Sets vis-a-vis $\Delta$-Sets. Konuralp J. Math. 2020;8(1):50-6.
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