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Year 2021, Issue: 36, 18 - 38, 30.09.2021

Mohammad Irshad Khodabocus , Noor-ul-hacq Sookıa

https://doi.org/10.53570/jnt.896345
Cited By: 7

Abstract

References

  • S. Ersoy, M. Bilgin, İ. İnce, Generalized Closed Set in Topological Spaces, Mathematica Moravica 19(1) (2015) 49-56.
  • S. Al Ghour, W. Zareer, Omega Open Sets in Generalized Topological Spaces, Journal of Nonlinear Sciences and Applications 9 (2016) 3010-3017.
  • P. Jeyanthi, P. Nalayini, M. Mocanu, g*λ_μ-Closed Sets and Generalized Topological Spaces, Boletim da Sociedade Paranaense de Matematica 34(1) (2016) 203-212.
  • I. Reilly, Generalized Closed Sets: A Survey of Recent Works, General and Geometric Topology and its Applications 1248 (2002) 1-11.
  • D. Saravanakumar, N. Kalaivani, G. S. S. Krishnan, On μ ̃-Open Sets in Generalized Topological Spaces, Malaya Journal of Matematik 3(3) (2015) 268-276.
  • B. K. Tyagi, Harsh V. S. Chauhan, On Generalized Closed Sets in a Generalized Topological Spaces, CUBO A Mathematical Journal 18(01) (2016) 27-45.
  • A. Danabalan, C. Santhi, A Class of Separation Axioms in Generalized Topology, Mathematical Journal of Interdisciplinary Sciences 4(2) (2016) 151-159.
  • Y. B. Jun, S. W. Jeong, H. J. Lee, J. W. Lee, Applications of Pre-Open Sets, Applied General Topology, Universidad Politecnica de Valencia 9(2) (2008) 213-228.
  • V. Pavlovic, A. S. Cvetkovic, On Generalized Topologies arising from Mappings, Vesnik, Universidad Politecnica de Valencia 38(3) (2012) 553-565.
  • O. Njastad, On Some Classes of Nearly Open Sets, Pacific Journal of Mathematics 15(3) (1965) 961-970.
  • D. Andrijevic, Semi-Pre-open Sets, Matematicki Vesnik 38(1) (1986) 24-32.
  • H. Ogata, Operations on Topological Spaces and Associated Topology, Mathematica Japonica 36 (1991) 175-184.
  • P. Jeyanthi, P. Nalayini, T. Noiri, Δ_μ-Sets and ∇_μ-Sets in Generalized Topological Spaces, Georgian Mathematical Journal 24(3) (2016) 403-407.
  • A. Csaszar, Generalized Topology, Generalized Continuity, Acta Mathematica Hungarica 96(4) (2002) 351-357.
  • N. Levine, Generalized Closed Set in Topological Spaces, Rendiconti del Circolo Matematico di Palermo 19 (1970) 89-96.
  • N. Levine, Semi-Open Sets and Semi-Continuity in Topological Spaces, American Mathematical Monthly 70 (1963) 19-41.
  • A. Csaszar, Remarks on Quasi-Topologies, Acta Mathematica Hungarica 119(1-2) (2008) 197-200.
  • A. Csaszar, Further Remarks on the Formula for γ-Interior, Acta Mathematica Hungarica 113(4) (2006) 325-332.
  • A. Csaszar, Generalized Open Sets in Generalized Topologies, Acta Mathematica Hungarica 106(1-2) (2005) 53-66.
  • A. Csaszar, Generalized Open Sets, Acta Mathematica Hungarica 75(1-2) (1997) 65-87.
  • K. Rajeshwari, T. D. Rayanagoudar, S. M. Patil, On Semi Generalized ωα-Closed Sets in Topological Spaces, Global Journal of Pure and Applied Mathematics 13(9) (2017) 5491-5503.
  • D. Andrijevic, On b-Open Sets, Matematicki Vesnik 48 (1996) 59-64.
  • S. Bayhan, A. Kanibir, I. L. Reilly, On Functions between Generalized Topological Spaces, Applied General Topology 14(2) (2013) 195-203.
  • P. Bhattacharyya, B.K. Lahiri, Semi-Generalized Closed Sets in Topology, Indian Journal of Mathematics 29 (1987) 376-382.
  • C. Boonpok, On Generalized Continuous Maps in Cech Closure Spaces, General Mathematics 19(3) (2011) 376-382.
  • J. Cao, M. Ganster, I. Reilly, On Generalized Closed sets, Topology and its Applications 123(1) (2002) 37-46.
  • J. Dontchev, T. Noiri, Quasi-Normal Spaces and πg-Closed Sets, Acta Mathematica Hungarica 89(3) (2000) 211-219.
  • J. Dontchev, On Generalizing Semi-Pre-open Sets, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 16 (1995) 35-48.
  • Y. Gnanambal, On Generalized Preregular Closed Sets in Topological Spaces, Indian Journal of Pure and Applied Mathematics 28 (1997) 351-360.
  • A. Gupta, R. D. Sarma, A Note on some Generalized Closure and Interior Operators in a Topological Space, Mathematics for Applications 6 (2017) 11-20.
  • R. A. Hosny, D. Al-Kadi, Types of Generalized Sets with Ideal, International Journal of Computer Applications 80(4) (2013) 11-14.
  • M. K. R. S. Veera Kumar, Between Closed Sets and g-Closed Sets, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 21 (2000) 1-19.
  • L. L. L. Lusanta, H. M. Rara, Generalized Star α-b-Separation Axioms in Bigeneralized Topological Spaces, Applied Mathematical Sciences 9(75) (2015) 3725-3737.
  • H. Maki, R. Devi, K. Balachandran, Associated Topologies of Generalized α-Closed Sets and α-Generalized Closed Sets, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 15 (1994) 51-63.
  • H. Maki, R. Devi, K. Balachandran, Generalized α-Closed Sets in Topology, Bulletin of Fukuoka University of Education Part III 42 (1993) 13-21.
  • A. S. Mashhour, I. A. Hasanein, S. N. E. Deeb, α-Continuous and α-Open Mappings, Acta Mathematica Hungarica 41(3-4) (1983) 213-218.
  • B. Roy, On a Type of Generalized Open Sets, Applied General Topology 12(2) (2011) 163-173.
  • P. Sundaram, M. Sheik John, On w-Closed Sets in Topology, Acta Ciencia Indica 4 (2000) 389-392.
  • M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius (2020) Reduit, Mauritius.
  • M. S. Sarsak, On some Properties of Generalized Open Sets in Generalized Topological Spaces, Demonstratio Mathematica XLVI (2) (2013) 415-427.
  • T. Noiri, Unified Characterizations for Modifications of R_0 and R_1 Topological Spaces, Circolo Matematico di Palermo 55(2) (2006) 29-42.
  • M. Caldas, S. Jafari, R. K. Saraf, Semi-θ-Open Sets and New Classes of Maps, Bulletin of the Iranian Mathematical Society 31(2) (2005) 37-52.
  • J. Dontchev, On Some Separation Axioms Associated with the α-Topology, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 18 (1997) 31-35.
  • M. Caldas, S. Jafari, On Some Applications of b-Open Sets in Topological Spaces, Kochi Journal of Mathematics 2 (2007) 11-19.

Theory of Generalized Sets in Generalized Topological Spaces

Year 2021, Issue: 36, 18 - 38, 30.09.2021

Mohammad Irshad Khodabocus , Noor-ul-hacq Sookıa

https://doi.org/10.53570/jnt.896345
Cited By: 7

Abstract

Several specific types of generalized sets (briefly, g-T_g-sets in generalized topological spaces (briefly, T_g-spaces have been defined and investigated for various purposes from time to time in the literature of T_g-spaces. Our recent research in the field of a new class of g-T_g-sets in T_g-spaces is reported herein as a starting point for more generalized classes. It is shown that the class of g-T_g-sets is a superclass of those whose elements are called open, closed, semi-open, semi-closed, pre-open, pre-closed, semi-pre-open, and semi-pre-closed sets in a T_g-space. A subclass of the T_g-subspace corresponds to the class of g-T_g-sets of a T_g-space. A class of g-T_g-sets of the Cartesian product of these T_g-spaces corresponds to the Cartesian product of a finite number of classes of g-T_g-sets, each of which belongs to a T_g-space. Diagrams establish the various relationships amongst the classes presented here and in the literature, and an ad hoc application supports the overall theory.

Keywords

Generalized topology Generalized topological space Generalized operations Generalized open sets Generalized closed sets

References

  • S. Ersoy, M. Bilgin, İ. İnce, Generalized Closed Set in Topological Spaces, Mathematica Moravica 19(1) (2015) 49-56.
  • S. Al Ghour, W. Zareer, Omega Open Sets in Generalized Topological Spaces, Journal of Nonlinear Sciences and Applications 9 (2016) 3010-3017.
  • P. Jeyanthi, P. Nalayini, M. Mocanu, g*λ_μ-Closed Sets and Generalized Topological Spaces, Boletim da Sociedade Paranaense de Matematica 34(1) (2016) 203-212.
  • I. Reilly, Generalized Closed Sets: A Survey of Recent Works, General and Geometric Topology and its Applications 1248 (2002) 1-11.
  • D. Saravanakumar, N. Kalaivani, G. S. S. Krishnan, On μ ̃-Open Sets in Generalized Topological Spaces, Malaya Journal of Matematik 3(3) (2015) 268-276.
  • B. K. Tyagi, Harsh V. S. Chauhan, On Generalized Closed Sets in a Generalized Topological Spaces, CUBO A Mathematical Journal 18(01) (2016) 27-45.
  • A. Danabalan, C. Santhi, A Class of Separation Axioms in Generalized Topology, Mathematical Journal of Interdisciplinary Sciences 4(2) (2016) 151-159.
  • Y. B. Jun, S. W. Jeong, H. J. Lee, J. W. Lee, Applications of Pre-Open Sets, Applied General Topology, Universidad Politecnica de Valencia 9(2) (2008) 213-228.
  • V. Pavlovic, A. S. Cvetkovic, On Generalized Topologies arising from Mappings, Vesnik, Universidad Politecnica de Valencia 38(3) (2012) 553-565.
  • O. Njastad, On Some Classes of Nearly Open Sets, Pacific Journal of Mathematics 15(3) (1965) 961-970.
  • D. Andrijevic, Semi-Pre-open Sets, Matematicki Vesnik 38(1) (1986) 24-32.
  • H. Ogata, Operations on Topological Spaces and Associated Topology, Mathematica Japonica 36 (1991) 175-184.
  • P. Jeyanthi, P. Nalayini, T. Noiri, Δ_μ-Sets and ∇_μ-Sets in Generalized Topological Spaces, Georgian Mathematical Journal 24(3) (2016) 403-407.
  • A. Csaszar, Generalized Topology, Generalized Continuity, Acta Mathematica Hungarica 96(4) (2002) 351-357.
  • N. Levine, Generalized Closed Set in Topological Spaces, Rendiconti del Circolo Matematico di Palermo 19 (1970) 89-96.
  • N. Levine, Semi-Open Sets and Semi-Continuity in Topological Spaces, American Mathematical Monthly 70 (1963) 19-41.
  • A. Csaszar, Remarks on Quasi-Topologies, Acta Mathematica Hungarica 119(1-2) (2008) 197-200.
  • A. Csaszar, Further Remarks on the Formula for γ-Interior, Acta Mathematica Hungarica 113(4) (2006) 325-332.
  • A. Csaszar, Generalized Open Sets in Generalized Topologies, Acta Mathematica Hungarica 106(1-2) (2005) 53-66.
  • A. Csaszar, Generalized Open Sets, Acta Mathematica Hungarica 75(1-2) (1997) 65-87.
  • K. Rajeshwari, T. D. Rayanagoudar, S. M. Patil, On Semi Generalized ωα-Closed Sets in Topological Spaces, Global Journal of Pure and Applied Mathematics 13(9) (2017) 5491-5503.
  • D. Andrijevic, On b-Open Sets, Matematicki Vesnik 48 (1996) 59-64.
  • S. Bayhan, A. Kanibir, I. L. Reilly, On Functions between Generalized Topological Spaces, Applied General Topology 14(2) (2013) 195-203.
  • P. Bhattacharyya, B.K. Lahiri, Semi-Generalized Closed Sets in Topology, Indian Journal of Mathematics 29 (1987) 376-382.
  • C. Boonpok, On Generalized Continuous Maps in Cech Closure Spaces, General Mathematics 19(3) (2011) 376-382.
  • J. Cao, M. Ganster, I. Reilly, On Generalized Closed sets, Topology and its Applications 123(1) (2002) 37-46.
  • J. Dontchev, T. Noiri, Quasi-Normal Spaces and πg-Closed Sets, Acta Mathematica Hungarica 89(3) (2000) 211-219.
  • J. Dontchev, On Generalizing Semi-Pre-open Sets, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 16 (1995) 35-48.
  • Y. Gnanambal, On Generalized Preregular Closed Sets in Topological Spaces, Indian Journal of Pure and Applied Mathematics 28 (1997) 351-360.
  • A. Gupta, R. D. Sarma, A Note on some Generalized Closure and Interior Operators in a Topological Space, Mathematics for Applications 6 (2017) 11-20.
  • R. A. Hosny, D. Al-Kadi, Types of Generalized Sets with Ideal, International Journal of Computer Applications 80(4) (2013) 11-14.
  • M. K. R. S. Veera Kumar, Between Closed Sets and g-Closed Sets, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 21 (2000) 1-19.
  • L. L. L. Lusanta, H. M. Rara, Generalized Star α-b-Separation Axioms in Bigeneralized Topological Spaces, Applied Mathematical Sciences 9(75) (2015) 3725-3737.
  • H. Maki, R. Devi, K. Balachandran, Associated Topologies of Generalized α-Closed Sets and α-Generalized Closed Sets, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 15 (1994) 51-63.
  • H. Maki, R. Devi, K. Balachandran, Generalized α-Closed Sets in Topology, Bulletin of Fukuoka University of Education Part III 42 (1993) 13-21.
  • A. S. Mashhour, I. A. Hasanein, S. N. E. Deeb, α-Continuous and α-Open Mappings, Acta Mathematica Hungarica 41(3-4) (1983) 213-218.
  • B. Roy, On a Type of Generalized Open Sets, Applied General Topology 12(2) (2011) 163-173.
  • P. Sundaram, M. Sheik John, On w-Closed Sets in Topology, Acta Ciencia Indica 4 (2000) 389-392.
  • M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius (2020) Reduit, Mauritius.
  • M. S. Sarsak, On some Properties of Generalized Open Sets in Generalized Topological Spaces, Demonstratio Mathematica XLVI (2) (2013) 415-427.
  • T. Noiri, Unified Characterizations for Modifications of R_0 and R_1 Topological Spaces, Circolo Matematico di Palermo 55(2) (2006) 29-42.
  • M. Caldas, S. Jafari, R. K. Saraf, Semi-θ-Open Sets and New Classes of Maps, Bulletin of the Iranian Mathematical Society 31(2) (2005) 37-52.
  • J. Dontchev, On Some Separation Axioms Associated with the α-Topology, Memoirs of the Faculty of the Science, Kochi University Series A Mathematics 18 (1997) 31-35.
  • M. Caldas, S. Jafari, On Some Applications of b-Open Sets in Topological Spaces, Kochi Journal of Mathematics 2 (2007) 11-19.
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Mohammad Irshad Khodabocus 0000-0003-2252-4342

Noor-ul-hacq Sookıa 0000-0002-3155-0473

Publication Date September 30, 2021
Submission Date March 13, 2021
Published in Issue Year 2021 Issue: 36

Cite

APA Khodabocus, M. I., & Sookıa, N.-u.-h. (2021). Theory of Generalized Sets in Generalized Topological Spaces. Journal of New Theory(36), 18-38. https://doi.org/10.53570/jnt.896345
AMA Khodabocus MI, Sookıa Nuh. Theory of Generalized Sets in Generalized Topological Spaces. JNT. September 2021;(36):18-38. doi:10.53570/jnt.896345
Chicago Khodabocus, Mohammad Irshad, and Noor-ul-hacq Sookıa. “Theory of Generalized Sets in Generalized Topological Spaces”. Journal of New Theory, no. 36 (September 2021): 18-38. https://doi.org/10.53570/jnt.896345.
EndNote Khodabocus MI, Sookıa N-u-h (September 1, 2021) Theory of Generalized Sets in Generalized Topological Spaces. Journal of New Theory 36 18–38.
IEEE M. I. Khodabocus and N.-u.-h. Sookıa, “Theory of Generalized Sets in Generalized Topological Spaces”, JNT, no. 36, pp. 18–38, September 2021, doi: 10.53570/jnt.896345.
ISNAD Khodabocus, Mohammad Irshad - Sookıa, Noor-ul-hacq. “Theory of Generalized Sets in Generalized Topological Spaces”. Journal of New Theory 36 (September 2021), 18-38. https://doi.org/10.53570/jnt.896345.
JAMA Khodabocus MI, Sookıa N-u-h. Theory of Generalized Sets in Generalized Topological Spaces. JNT. 2021;:18–38.
MLA Khodabocus, Mohammad Irshad and Noor-ul-hacq Sookıa. “Theory of Generalized Sets in Generalized Topological Spaces”. Journal of New Theory, no. 36, 2021, pp. 18-38, doi:10.53570/jnt.896345.
Vancouver Khodabocus MI, Sookıa N-u-h. Theory of Generalized Sets in Generalized Topological Spaces. JNT. 2021(36):18-3.

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