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Year 2023, Volume: 5 Issue: 1, 37 - 62, 18.07.2023
https://doi.org/10.47086/pims.1214064

Abstract

References

  • Reference1 M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory 36 (2021) 18-38.
  • Reference2 M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R´eduit, Mauritius (2020) 1-311 (i.-xxxvi.).
  • Reference3 S. -M. Jung and D. Nam, Some Properties of Interior and Closure in General Topology, Mathematics (MDPI Journal) 7(624) (2019) 1-10.
  • Reference4 Y. Lei and J. Zhang, Generalizing Topological Set Operators, Electronic Notes in Theoretical Science 345 (2019) 63–76.
  • Reference5 A. Gupta and R. D. Sarma, A Note on some Generalized Closure and Interior Operators in a Topological Space, Math. Appl. 6 (2017) 11-20.
  • Reference6 R. Rajendiran and M. Thamilselvan, Properties of g∗s∗-Closure, g∗s∗-Interior and g∗s∗- Derived Sets in Topological Spaces, Applied Mathematical Sciences 8(140) (2014) 6969–6978.
  • Reference7 B. K. Tyagi and R. Choudhary, On Generalized Closure Operators in Generalized Topological Spaces, International Journal of Computer Applications 82(15) (2013) 1-5.
  • Reference8 C. Cao and B. Wang and W. Wang, Generalized Topologies, Generalized Neighborhood Systems, and Generalized Interior Operators, Acta Math. Hungar. 132(4) (2011) 310-315.
  • Reference9 V. Pankajam, On the Properties of δ-Interior and δ-Closure in Generalized Topological Spaces, International Journal for Research in Mathematical Archive 2(8) (2011) 1321-1332.
  • Reference10 B. J. Gardner and M. Jackson, The Kuratowski Closure-Complement Theorem, New Zealand Journal of Mathematics 38 (2008) 9-44.
  • Reference11 ´A. Cs´asz´ar, Further Remarks on the Formula for γ-Interior, Acta Math. Hungar. 113(4) (2006) 325-332.
  • Reference12 ´A. Cs´asz´ar, On the γ-Interior and γ-Closure of a Set, Acta Math. Hungar. 80 (1998) 89-93.
  • Reference13 H. Ogata, Operations on Topological Spaces and Associated Topology, Math. Japonica 36 (1991) 175-184.
  • Reference14 I. Z. Kleiner, Closure and Boundary Operators in Topological Spaces, Ukr Math. J. 29 (1977) 295-296.
  • Reference15 F. R. Harvey, The Derived Set Operator, The American Mathematical Monthly 70(10) (1963) 1085-1086.
  • Reference16 D. H. Staley, On the Commutivity of the Boundary and Interior Operators in a Topological Space, The Ohio Journal of Science 68(2):84 (1968).
  • Reference17 N. Levine, On the Commutivity of the Closure and Interior Operators in Topological Spaces, Amer. Math. Monthly 68(5) (1961) 474-477.
  • Reference18 C. Kuratowski, Sur l’Op´eration A¯ de l’Analyse Situs, Fund. Math. 3 (1922) 182-199.
  • Reference19 M. I. Khodabocus and N. -Ul. -H. Sookia, Generalized Topological Operator Theory in Generalized Topological Spaces: Part I. Generalized Interior and Generalized Closure, Proceedings of International mathematical Sciences 5(1) (2023) 6-36.
  • Reference20 M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics 5(1) (2022) 1-23.
  • Reference21 ´A. Cs´asz´ar, Generalized Open Sets in Generalized Topologies, Acta Math. Hungar. 106(1-2) (2005) 53-66.
  • Reference22 V. Pavlovi´c and A. S. Cvetkovi´c, On Generalized Topologies arising from Mappings, Vesnik 38(3) (2012) 553-565.
  • Reference23 ´A. Cs´asz´ar, Remarks on Quasi-Topologies, Acta Math. Hungar. 119(1-2) (2008) 197-200.

Generalized Topological Operator Theory in Generalized Topological Spaces: Part II. Generalized Interior and Generalized Closure

Year 2023, Volume: 5 Issue: 1, 37 - 62, 18.07.2023
https://doi.org/10.47086/pims.1214064

Abstract

In a recent paper (Cf. [19]), we have presented the definitions and the essential properties of the generalized topological operators
g-Int_g, g-Cl_g : P (Ω) −→ P (Ω) (g-T_g-interior and g-T_g-closure operators) in a generalized topological space T_g = (Ω, T_g) (T_g-space). Principally, we have
shown that (g-Int_g , g-Cl_g) : P (Ω) × P (Ω) −→ P (Ω) × P (Ω) is (Ω, ∅)-grounded, (expansive, non-expansive), (idempotent, idempotent) and (∩, ∪)-additive. We have also shown that g-Int_g : P (Ω) −→ P (Ω) is finer (or, larger, stronger) than int_g : P (Ω) −→ P (Ω) and g-Cl_g : P (Ω) −→ P (Ω) is coarser (or, smaller, weaker) than
cl_g : P (Ω) −→ P (Ω). In this paper, we study the commutativity of g-Int_g , g-Cl_g : P (Ω) −→ P (Ω) and T_g-sets having some (g-Int_g, g-Cl_g)-based properties
(g-P_g, g-Q_g-properties) in T_g-spaces. The main results of the study are: The g-T_g-operators g-Int_g, g-Cl_g : P (Ω) −→ P (Ω) are duals and g-P_g-property is preserved under their g-T_g-operations. A T_g-set having g-P_g-property is equivalent to the T_g-set or its complement having g-Q_g-property.
The g-Q_g-property is preserved under the set-theoretic ∪-operation and g-P_g-property is preserved under the set-theoretic {∪, ∩, C}-operations. Finally, a T_g-set having {g-P_g , g-Q_g}-property also has {P_g , Q_g }-property.

References

  • Reference1 M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory 36 (2021) 18-38.
  • Reference2 M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R´eduit, Mauritius (2020) 1-311 (i.-xxxvi.).
  • Reference3 S. -M. Jung and D. Nam, Some Properties of Interior and Closure in General Topology, Mathematics (MDPI Journal) 7(624) (2019) 1-10.
  • Reference4 Y. Lei and J. Zhang, Generalizing Topological Set Operators, Electronic Notes in Theoretical Science 345 (2019) 63–76.
  • Reference5 A. Gupta and R. D. Sarma, A Note on some Generalized Closure and Interior Operators in a Topological Space, Math. Appl. 6 (2017) 11-20.
  • Reference6 R. Rajendiran and M. Thamilselvan, Properties of g∗s∗-Closure, g∗s∗-Interior and g∗s∗- Derived Sets in Topological Spaces, Applied Mathematical Sciences 8(140) (2014) 6969–6978.
  • Reference7 B. K. Tyagi and R. Choudhary, On Generalized Closure Operators in Generalized Topological Spaces, International Journal of Computer Applications 82(15) (2013) 1-5.
  • Reference8 C. Cao and B. Wang and W. Wang, Generalized Topologies, Generalized Neighborhood Systems, and Generalized Interior Operators, Acta Math. Hungar. 132(4) (2011) 310-315.
  • Reference9 V. Pankajam, On the Properties of δ-Interior and δ-Closure in Generalized Topological Spaces, International Journal for Research in Mathematical Archive 2(8) (2011) 1321-1332.
  • Reference10 B. J. Gardner and M. Jackson, The Kuratowski Closure-Complement Theorem, New Zealand Journal of Mathematics 38 (2008) 9-44.
  • Reference11 ´A. Cs´asz´ar, Further Remarks on the Formula for γ-Interior, Acta Math. Hungar. 113(4) (2006) 325-332.
  • Reference12 ´A. Cs´asz´ar, On the γ-Interior and γ-Closure of a Set, Acta Math. Hungar. 80 (1998) 89-93.
  • Reference13 H. Ogata, Operations on Topological Spaces and Associated Topology, Math. Japonica 36 (1991) 175-184.
  • Reference14 I. Z. Kleiner, Closure and Boundary Operators in Topological Spaces, Ukr Math. J. 29 (1977) 295-296.
  • Reference15 F. R. Harvey, The Derived Set Operator, The American Mathematical Monthly 70(10) (1963) 1085-1086.
  • Reference16 D. H. Staley, On the Commutivity of the Boundary and Interior Operators in a Topological Space, The Ohio Journal of Science 68(2):84 (1968).
  • Reference17 N. Levine, On the Commutivity of the Closure and Interior Operators in Topological Spaces, Amer. Math. Monthly 68(5) (1961) 474-477.
  • Reference18 C. Kuratowski, Sur l’Op´eration A¯ de l’Analyse Situs, Fund. Math. 3 (1922) 182-199.
  • Reference19 M. I. Khodabocus and N. -Ul. -H. Sookia, Generalized Topological Operator Theory in Generalized Topological Spaces: Part I. Generalized Interior and Generalized Closure, Proceedings of International mathematical Sciences 5(1) (2023) 6-36.
  • Reference20 M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics 5(1) (2022) 1-23.
  • Reference21 ´A. Cs´asz´ar, Generalized Open Sets in Generalized Topologies, Acta Math. Hungar. 106(1-2) (2005) 53-66.
  • Reference22 V. Pavlovi´c and A. S. Cvetkovi´c, On Generalized Topologies arising from Mappings, Vesnik 38(3) (2012) 553-565.
  • Reference23 ´A. Cs´asz´ar, Remarks on Quasi-Topologies, Acta Math. Hungar. 119(1-2) (2008) 197-200.
There are 23 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Articles
Authors

Mohammad Irshad Khodabocus 0000-0003-2252-4342

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

Early Pub Date July 17, 2023
Publication Date July 18, 2023
Acceptance Date May 2, 2023
Published in Issue Year 2023 Volume: 5 Issue: 1

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