On generalized open sets

Year 2018, Volume: 47 Issue: 6, 1438 - 1446, 12.12.2018

Sabir Hussain

Abstract

We introduce and explore generalized open sets namely $\gamma$-pre-open and $\gamma$-b-open sets in topological spaces. We also explore and investigate the properties and characterizations  of $\gamma$-b-continuous function, $\gamma$-b-irresolute function and  $\gamma$-b-open(closed)
functions. Examples and counter examples are also provided for the existence of defined concepts.

Keywords

$\gamma$-open(closed), $\gamma$-interior(closure), $\gamma$-semi-open(closed), $\gamma$-semi-closure(interior), $\gamma$-pre-open(closed), $\gamma$-b-open(closed), $\gamma$-b-continuous function, $\gamma$-b-irresolute function, $\gamma$-b-open(closed) function

References

• Ahmad, B., Hussain, S., Properties of $\gamma$-Operations on Topological Spaces, Aligarh Bull. Math. 22(1), 45-51, 2003.
• Ahmad, B., Hussain, S., $\gamma$-Convergence in Topological Space, Southeast Asian Bull. Math. 29, 832-842, 2005.
• Ahmad, B., Hussain, S., $\gamma^*$-Regular and $\gamma$-Normal Space, Math. Today. 22(1), 37-44, 2006.
• Ahmad, B., Hussain, S., On $\gamma$-s-Closed Subspaces, Far East Jr. Math. Sci. 31(2), 279-291, 2008.
• Ahmad, B., Hussain, S., $\gamma$-Semi-Open Sets in Topological Spaces II, Southeast Asian Bull. Math. 34(5), 997-1008, 2010.
• Ahmad, B., Hussain, S., Noiri, T., On Some Mappings in Topological Space, Eur. J. Pure Appl. Math. 1, 22-29, 2008.
• Hussain, S., Ahmad, B., On $\gamma$-s-Closed Spaces, Sci. Magna Jr. 3(4), 89-93, 2007.
• Hussain, S., Ahmad, B., On $\gamma$-s-Regular Spaces and Almost $\gamma$-s-Continuous Functions, Lobackevskii. J. Math. 30(4), 263-268, 2009.
• Hussain, S., Ahmad, B., Applications of Minimal $\gamma$-Semi-Open Sets, Fasciculli Mathematici 46, 65-76, 2011.
• Hussain, S., Ahmad, B., Noiri, T., $\gamma$-Semi-Open Sets in Topological Spaces, Asian Eur. J. Math. 3(3), 427-433, 2010.
• Hussain, S., Ahmad, B., Noiri, T., On $\gamma$-Semi Continuous Functions, Punjab University Jr. Math. 42, 57-65, 2010.
• Hussain, S., Alghamdi, M. A., Some Properties of Weak form of $\gamma$-Semi-Open Sets, Tamkang Jr. Math. 42, 57-65 2010.
• Kasahara, S., Operation-Compact Spaces, Math. Japon. 24, 97-105, 1979.
• Levine, N., Semi-Open Sets and Semi Continuity in Topological Spaces, Amer. Math. Monthly 70, 36-41, 1963.
• Levine, N., Generalized Closed Sets in Topological Spaces, Rend. Ser. Math. Palerno. 2(19), 89-96, 1970.
• Najastad, O., On Some Classes of Nearly Open Sets, Pacific J. Math. 15, 961-970, 1965.
• Ogata, H., Operations on Topological Spaces and Associated Topology, Math. Japon. 36(1), 175-184, 1991.
• Rehman, F. U., Ahmad, B., Operations on Topological Spaces I, Math. Today 10, 29-36, 1992.
• Umehara, J., Maki, H., Noiri, T., Bi Operations on Topological Spaces and Some Separation Axioms, Mem Fac. Sci. Kochi University Ser. A. Math. 13, 45-59, 1992.
• Umehara, J., Maki, H., Noiri, T., A Certain Bi Operations On Topological Spaces, Mem Fac. Sci. Kochi University Ser. A. Math. 15, 41-49, 1994.