# On Weak and Strong Forms of Beta-open Sets

## Abstract

In this paper, new classes of sets in general topology called asupra--open (closed)(an infra--open (closed)) set are introduced. Using new concepts, the fundamental properties and special results are highlighted.

The relations between supra--open (closed)(an infra--open (closed)) set and other topological sets are investigated. Moreover, counter-examples are given to show that the converse of these relations in Diagram 1 need not be true, in general. Finally, some special theorems are introduced by adding condition to achieve the converse relations in Diagram 1.

## References

[1] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud. Beta-open sets and Beta-continuous mapping. Bull. Fac. Sci. Assiut Univ., 12(1)(1983), 77-90.

[2] W. Dunham. A New Closure Operator for non-T1 topology. Kyuungpook Math. J., 22(1982), 55-60.

[3] MS. El Naschie. On the uncertainty of Cantorian geometry and the two-slit experiment, Chaos, Solitons and Fractals, 9(3)(1998), 517-529.

[4] MS. El Naschie. Quantum gravity from descriptive set theory. Chaos, Solitons and Fractals, 19(5)(2004), 1339-1344.

[5] MS. El Naschie. Quantum gravity, Clifford algebras, fuzzy set theory and the fundamental constants of nature. Chaos, Solitons and Fractals, 20(3)(2004), 437-450.

[6] MS. El Naschie. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int. J. Nonlinear Sci. Numer Simul, 6(2)(2005), 95-98.

[7] MS. El Naschie. Topics in the mathematical physics of E-infinity theory. Chaos, Solitons and Fractals, 30(3)(2006), 656-663.

[8] Hakeem A. Othman and Md. Hanif Page. ON An Infra-Alfa-Open Sets. Global J. Math. Anal., 4(3)(2016), 12-16.

[9] N. Levine. Semi open sets and semi-continuouity in topological space. Amer. Math. Monthly, 70(1)(1981), 36-41.

[10] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb. On Precontinuous and weak precontinuous Mappings. Proc. Math. Phys. Soc. Egypt, 53(1982), 47-53.

[11] O. Njastad. Some Classes of Nearly Open sets. Pacific J. Math., 15(3)(1965), 961-970.

[12] S. Pious Missier and P. Anbarasi Rodrigo. Some Notions of Nearly Open Sets in Toological Spaces. Inter. J. Math. Archive, 4(12)(2013), 12-18.

[13] A. Robert and S. Pious Missier. A New Class of Nearly Open Sets. Inter. J. Math. Archive, 3(7)(2012), 2575-2582.

[14] T. Selvi and A. Punitha Dharani. Some new class of nearly closed and open sets. Asian J. Current Engineering and Maths., 1(5)(2012), 305-307.

[2] W. Dunham. A New Closure Operator for non-T1 topology. Kyuungpook Math. J., 22(1982), 55-60.

[3] MS. El Naschie. On the uncertainty of Cantorian geometry and the two-slit experiment, Chaos, Solitons and Fractals, 9(3)(1998), 517-529.

[4] MS. El Naschie. Quantum gravity from descriptive set theory. Chaos, Solitons and Fractals, 19(5)(2004), 1339-1344.

[5] MS. El Naschie. Quantum gravity, Clifford algebras, fuzzy set theory and the fundamental constants of nature. Chaos, Solitons and Fractals, 20(3)(2004), 437-450.

[6] MS. El Naschie. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int. J. Nonlinear Sci. Numer Simul, 6(2)(2005), 95-98.

[7] MS. El Naschie. Topics in the mathematical physics of E-infinity theory. Chaos, Solitons and Fractals, 30(3)(2006), 656-663.

[8] Hakeem A. Othman and Md. Hanif Page. ON An Infra-Alfa-Open Sets. Global J. Math. Anal., 4(3)(2016), 12-16.

[9] N. Levine. Semi open sets and semi-continuouity in topological space. Amer. Math. Monthly, 70(1)(1981), 36-41.

[10] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb. On Precontinuous and weak precontinuous Mappings. Proc. Math. Phys. Soc. Egypt, 53(1982), 47-53.

[11] O. Njastad. Some Classes of Nearly Open sets. Pacific J. Math., 15(3)(1965), 961-970.

[12] S. Pious Missier and P. Anbarasi Rodrigo. Some Notions of Nearly Open Sets in Toological Spaces. Inter. J. Math. Archive, 4(12)(2013), 12-18.

[13] A. Robert and S. Pious Missier. A New Class of Nearly Open Sets. Inter. J. Math. Archive, 3(7)(2012), 2575-2582.

[14] T. Selvi and A. Punitha Dharani. Some new class of nearly closed and open sets. Asian J. Current Engineering and Maths., 1(5)(2012), 305-307.

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## Published

2016-12-26

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## Section

Чланци