THEORY OF GENERALIZED MAPS IN GENERALIZED TOPOLOGICAL SPACES

Öz

It is the intent of this paper to present the definitions of new families of generalized maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps) and generalized continuous ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous), generalized irresolute (briefly, $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute) and generalized homeomorphism ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-homeomorphism) maps between generalized topological spaces ($\mathcal{T}_{\mathfrak{g}}$-spaces) in terms of generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets) and study their fundamental properties. For each type of such $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps, necessary and sufficient conditions are determined. These classes are found to be superclasses of those whose elements are defined in terms of the classes of open, closed sets, semi-open, semi-closed sets, preopen, preclosed sets, and semi-preopen, semi-preclosed sets. The investigation reveals that the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous maps is a superclass of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps and both classes of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps are subclasses of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps. To render this work well-founded, diagrams are inserted to exhibit several relationships amongst the $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps studied in this paper and in the literature, and an illustrative application is given to support some theoretical facts.

Anahtar Kelimeler

• Generalized topological space ($\mathcal{T}_{\mathfrak{g}}$-spaces)
• generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets)
• generalized maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps)
• generalized continuous maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous maps)
• generalized irresolute maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps)
• generalized homeomorphism maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-homeomorphism maps)

THEORY OF GENERALIZED MAPS IN GENERALIZED TOPOLOGICAL SPACES

Öz

It is the intent of this paper to present the definitions of new families of generalized maps (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps}) and generalized continuous (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous}), generalized irresolute (briefly, \emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute}) and generalized homeomorphism (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-homeomorphism}) maps between generalized topological spaces (\emph{$\mathcal{T}_{\mathfrak{g}}$-spaces}) in terms of generalized sets (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets}) and study their fundamental properties. For each type of such $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps, necessary and sufficient conditions are determined. These classes are found to be superclasses of those whose elements are defined in terms of the classes of open, closed sets, semi-open, semi-closed sets, preopen, preclosed sets, and semi-preopen, semi-preclosed sets. The investigation reveals that the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous maps is a superclass of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps and both classes of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps are subclasses of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps. To render this work well-founded, diagrams are inserted to exhibit several relationships amongst the $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps studied in this paper and in the literature, and an illustrative application is given to support some theoretical facts.

Anahtar Kelimeler

• Generalized topological space ($\mathcal{T}_{\mathfrak{g}}$-spaces)
• generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets)
• generalized maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps)
• generalized continuous maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous maps)
• generalized irresolute maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps)
• generalized homeomorphism maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-homeomorphism maps)

Kaynakça

1. R. Arens and J. Dugundji, Topologies for Function Spaces, Pacific Journal of Mathematics, Vol.1, pp.5-31 (1951).
2. A. V. Arhangel’skii, Mappings and Spaces, Russian Mathematical Surveys, Vol.21, N.4, pp.115-162 (1966).
3. M. C. Cueva, Semi-Generalized Maps in Topological Spaces, Portugaliae Mathematica, Vol.52, N.4, pp.399-407 (1995).
4. N. Levine, A Decomposition of Continuity in Topological Spaces, American Mathematical Monthly, Vol.68, pp.44-46 (1961).
5. S. Bayhan, A. Kanıbir and I. L. Reilly, On Functions between Generalized Topological Spaces, Applied General Topology, Vol.14, N.2, pp.195-203 (2013).
6. W. K. Min, Generalized Continuous Functions Defined by Generalized Open Sets on Generalized Topological Spaces, Acta Mathematica Hungarica, Vol.128, N.4, pp.299-306 (2010).
7. X. Ge, J. Gong and I. Reilly, Some Characterizations of Mappings on Generalized Topological Spaces, New Zealand Journal of Mathematics, Vol.46, pp.73-81 (2016).
8. K. Kannan, N. Nagaveni and S. Saranya, On β̂g-Continuous and β̂g-Irresolute Maps in Topological Spaces, Procedia Computer Science, Vol.21, N.4, pp.115-162 (1966).

9. E. Korczak-Kubiak and R. J. Pawlak, On Semi-Open Sets and Mutual Correspondence Between Properties of Functions Considered with Respect to Different Topological Structures, Tatra Mountains Mathematical Publications, Vol.65, N.1, pp.119-134 (2016).
10. A. H. Zakari, gm-Continuity on Generalized Topology and Minimal Structures Spaces, Journal of the Association of Arab Universities for Basic and Applied Sciences, Vol.20, pp.78-83 (2018).
11. F. M. V. Valenzuela and H. M. Rara, µ-rgb-Sets in a Generalized Topological Space, International Journal of Mathematical Analysis, Vol.8, N.36, pp.1791-1797 (2014).
12. R. Messaoud and A. Missaoui, Properties of the Contra Semi-Continuity Functions and σ-ζµ-Sets in Generalized Topological Spaces, Mathematics for Applications, Vol.4, pp.123-128 (2015).
13. P. Montagantirud and W. Thaikua, Continuities on Generalised Topological Spaces via Hereditary Classes, Bulletin of the Australian Mathematical Society, Vol.97, N.2, pp.320-330 (2018).
14. M. L. Thivagar, I. L. Reilly, M. A. Dasan and V. Ramesh, Generalized Open Sets in Grill N-Topology, Applied General Topology, Vol.18, N.2, pp.289-299 (2017).
15. S. G. Crossley and S. K. Hildebrand, Semi-Topological Properties, Fundamenta Mathematicae, Vol.74, N.3, pp.233-254 (1972).
16. G. L. Garg and D. Sivaraj, Generalized Continuous Functions Defined by Generalized Open Sets on Generalized Topological Spaces, Soochow Journal of Mathematics, Vol.14, N.1, pp.51-55 (1988).
17. S. Jafari and N. Rajesh, Weak and Strong Form of g̃-Irresolute Functions, Filomat, Vol.21, N.2, pp.199-209 (2007).
18. S. Jafari and T. Noiri, Contra-α-Continuous Functions between Topological Spaces, Iranian International Journal of Science, Vol.2, N.2, pp.153-167 (2001).
19. W. K. Min, On g-α-Irresolute Functions, Acta Mathematica Hungarica, Vol.130, N.4, pp.385-389 (2011).
20. G. H. Moore, The Evolution of the Concept of Homeomorphism, Historia Mathematica, Vol.34, N.3, pp.333-343 (2007).
21. T. Noiri and V. Popa, A Generalization of Some Forms of g-Irresolute Functions, European Journal of Pure and Applied Mathematics, Vol.2, N.4, pp.473-493 (2009).
22. P. L. Powar and K. Rajak, Some New Concepts of Continuity in Generalized Topological Space, International Journal of Computer Applications, Vol.38, N.5, pp.12-17 (2011).
23. S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, (1970).
24. A. S. Mashhour, I. A. Hasanein and S. N. E. Deeb, α-Continuous and α-Open Mappings, Acta Mathematica Hungarica, Vol.41, N.3-4, pp.213-218 (1983).
25. C. Boonpok, On Generalized Continuous Maps in Čech Closure Spaces, General Mathematics, Vol.19, N.3, pp.3-10 (2011).
26. I. Zvina, Introduction to Generalized Topological Spaces, Applied General Topology, Vol.12, N.1, pp.49-66 (2011).
27. M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory, Vol.36, pp.18-38 (2021).
28. M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, Ph.D. Thesis, University of Mauritius, (2020).
29. M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics, Vol.5, N.1, pp.1-23 (2022).
30. M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties, Fundamentals of Contemporary Mathematical Sciences, Vol.3, N.1, pp.26-45 (2022).
31. Á. Császár, Remarks on Quasi-Topologies, Acta Mathematica Hungarica, Vol.119, N.1-2, pp.197-200 (2008).
32. Á. Császár, Generalized Open Sets in Generalized Topologies, Acta Mathematica Hungarica, Vol.106, N.1-2, pp.53-66 (2005).
33. V. Pavlović and A. S. Cvetković, On Generalized Topologies arising from Mappings, Matematički Vesnik, Vol.38, N.3, pp.553-565 (2012).
34. M. Caldas, S. Jafari and R. K. Saraf, Semi-θ-Open Sets and New Classes of Maps, Bulletin of the Iranian Mathematical Society, Vol.31, N.2, pp.37-52 (2005).
35. J. Dontchev, On Some Separation Axioms Associated with the α-Topology, Memoirs of the Faculty of Science, Kochi University Series A Mathematics, Vol.18, pp.31-35 (1997).
36. Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee, Applications of Pre-Open Sets, Applied General Topology, Vol.9, N.2, pp.213-228 (2008).
37. B. K. Tyagi and V. S. Chauhan Harsh, On Generalized Closed Sets in a Generalized Topological Spaces, CUBO A Mathematical Journal, Vol.18, N.1, pp.27-45 (2016).
38. T. Noiri, A. Al-Omari and M. S. M. Noorani, Weakly b-Open Functions, Mathematica Balkanica, Vol.23, N.1-2, pp.1-13 (2009).