On gamma-Ps-operations in topological spaces

Topology is one of the focus areas in mathematics. Recently, topology has become an important component in applied mathematics due to its vast applications in understanding real life problems. The basic concept of topological space (X, Ƭ) deals with open sets. Operations on Ƭ have been investigate...

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Main Author:	Asaad, Baravan Abdulmuhsen
Format:	Thesis
Language:	eng eng
Published:	2015
Subjects:	QA Mathematics
Online Access:	https://etd.uum.edu.my/5800/1/s93362_02.pdf https://etd.uum.edu.my/5800/2/s93362_01.pdf
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id	my-uum-etd.5800
record_format	uketd_dc
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng eng
advisor	Ahmad, Nazihah Omar, Zurni
topic	QA Mathematics
spellingShingle	QA Mathematics Asaad, Baravan Abdulmuhsen On gamma-Ps-operations in topological spaces
description	Topology is one of the focus areas in mathematics. Recently, topology has become an important component in applied mathematics due to its vast applications in understanding real life problems. The basic concept of topological space (X, Ƭ) deals with open sets. Operations on Ƭ have been investigated by numerous researchers. Among these operations are γ-open, γ-preopen, γ-semiopen, γ-b-open, γ-β-open and α-γ-open which involve Ƭγ-Ps-interior and Ƭγ-Ps-closure. However, no one has attempted to define new class of set using operation γ on the topology Ƭ by combining the existing operations. This study, therefore, aims to define new classes of sets, construct new classes of functions, and introduce new types of separation axioms and spaces using the concept of γ-open sets. The new classes developed are γ-regular-open and γ-Ps-open sets. By applying γ-Ps-open sets and their complements, the notions of Ƭγ-Ps-closure, Ƭγ-Ps-interior, Ƭγ-Ps-derived set and Ƭγ-Ps-boundary of a set are established. The notions of γ-Ps-open and Ƭγ-Ps-closure sets are then used to define a new class of γ-Ps-open sets called γ-Ps-generalised closed sets. Moreover, several new classes of functions called γ-Ps-continuous, (γ,β)-Ps-continuous and (γ,β)-Ps-irresolute functions in term of γ-Ps-open sets are introduced. Furthermore, other types of γ-Ps-functions such as β-Ps-open and (γ,β)-Ps-open are constructed. In addition, some new classes of γ-Ps-separation axioms are established by using γ-Ps-open and its complement as well as γ-Ps-generalised closed sets. The relationships and properties of each class of sets, γ-Ps-functions and γ-Ps-separation axioms are also established. In conclusion, this study has succeeded in defining new classes of sets using operation γ on the topology Ƭ .
format	Thesis
qualification_name	Ph.D.
qualification_level	Doctorate
author	Asaad, Baravan Abdulmuhsen
author_facet	Asaad, Baravan Abdulmuhsen
author_sort	Asaad, Baravan Abdulmuhsen
title	On gamma-Ps-operations in topological spaces
title_short	On gamma-Ps-operations in topological spaces
title_full	On gamma-Ps-operations in topological spaces
title_fullStr	On gamma-Ps-operations in topological spaces
title_full_unstemmed	On gamma-Ps-operations in topological spaces
title_sort	on gamma-ps-operations in topological spaces
granting_institution	Universiti Utara Malaysia
granting_department	Awang Had Salleh Graduate School of Arts & Sciences
publishDate	2015
url	https://etd.uum.edu.my/5800/1/s93362_02.pdf https://etd.uum.edu.my/5800/2/s93362_01.pdf
_version_	1747827984715218944
spelling	my-uum-etd.58002021-03-18T04:01:20Z On gamma-Ps-operations in topological spaces 2015 Asaad, Baravan Abdulmuhsen Ahmad, Nazihah Omar, Zurni Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA Mathematics Topology is one of the focus areas in mathematics. Recently, topology has become an important component in applied mathematics due to its vast applications in understanding real life problems. The basic concept of topological space (X, Ƭ) deals with open sets. Operations on Ƭ have been investigated by numerous researchers. Among these operations are γ-open, γ-preopen, γ-semiopen, γ-b-open, γ-β-open and α-γ-open which involve Ƭγ-Ps-interior and Ƭγ-Ps-closure. However, no one has attempted to define new class of set using operation γ on the topology Ƭ by combining the existing operations. This study, therefore, aims to define new classes of sets, construct new classes of functions, and introduce new types of separation axioms and spaces using the concept of γ-open sets. The new classes developed are γ-regular-open and γ-Ps-open sets. By applying γ-Ps-open sets and their complements, the notions of Ƭγ-Ps-closure, Ƭγ-Ps-interior, Ƭγ-Ps-derived set and Ƭγ-Ps-boundary of a set are established. The notions of γ-Ps-open and Ƭγ-Ps-closure sets are then used to define a new class of γ-Ps-open sets called γ-Ps-generalised closed sets. Moreover, several new classes of functions called γ-Ps-continuous, (γ,β)-Ps-continuous and (γ,β)-Ps-irresolute functions in term of γ-Ps-open sets are introduced. Furthermore, other types of γ-Ps-functions such as β-Ps-open and (γ,β)-Ps-open are constructed. In addition, some new classes of γ-Ps-separation axioms are established by using γ-Ps-open and its complement as well as γ-Ps-generalised closed sets. The relationships and properties of each class of sets, γ-Ps-functions and γ-Ps-separation axioms are also established. In conclusion, this study has succeeded in defining new classes of sets using operation γ on the topology Ƭ . 2015 Thesis https://etd.uum.edu.my/5800/ https://etd.uum.edu.my/5800/1/s93362_02.pdf text eng public https://etd.uum.edu.my/5800/2/s93362_01.pdf text eng public Ph.D. doctoral Universiti Utara Malaysia Abd El-Monsef M. E., El-Deeb S. N. & Mahmoud R. A. (1983). β-open sets and β-continuous mappings, Bull. Fac. Sci. Assuit. Univ., 12(1), 1-18. Abd El-Monsef M. E., Mahmoud R. A. & Lashin E. R. (1986). β-closure and β-interior, J. Fac. Ed. Ain. Shams. Univ., 10, 235-245. Andrijevic D. (1996). On b-open sets, Matematicki Vesnik, 48(1-2), 59-64. Andrijevic D. (1986). Semi-preopen sets, Matematicki Vesnik, 38, 24-32. Arya S. P. & Gupta R. (1974). On strongly continuous functions, Kyungpook Mathematical Journal, 14, 131-143. Basu, C. K., Afsan, B. M. U. & Ghosh, M. 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(2003). A new class of semi open sets in a topological space, Proc. NCMCM, Allied Publishers, New Delhi, 305-311. Krishnan G. S. S. & Balachandran K. (2006a). On a class of γ-preopen sets in a topological space, East Asian Math. J., 22(2), 131-149. Krishnan G. S. S. & Balachandran K. (2006b). On γ-semiopen sets in topological spaces, Bulletin Calcutta Mathematical Society, 98(6), 517-530. Krishnan G. S. S., Ganster M. & Balachandran K. (2007). Operation approaches on semiopen sets and applications, Kochi Journal of Mathematics, 2, 21-33. Levine, N. (1963). Semi-open sets and semi-continuity in topological spaces, American Mathematical Monthly, 70(1), 36-41. Long P. E. (1986). An Introduction to General Topology, Charles E. Merrill Publishing Company. Mashhour, A. S., Abd El-Monsef, M. E. & El-Deeb, S. N. (1982). On pre-continuous and week precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53, 47-53. Njastad O. (1965). On some classes of nearly open sets, Pacific Journal of Mathematics, 15(3), 961-970. Ogata, H. (1991). Operation on topological spaces and associated topology, Mathematica Japonica, 36(1), 175-184. Reilly I. L. & Vamanmurthy M. K. (1985). On α-continuity in topological spaces, Acta Mathematica Hungarica, 45(1-2), 27-32. Steen L. A. & Seebach J. A. (1978). Counterexamples in Topology, New York Heidelberg Berlin: Springer Verlag. Tahiliani S. (2011). Operation approach to β-open sets and applications, Mathematical Communications, 16, 577-591. Van An T., Cuong D. X. & Maki H. (2008). On operation-preopen sets in topological spaces, Scientiae Mathematicae Japonicae Online, e-2008, 241-260.

On gamma-Ps-operations in topological spaces

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