On wgrα - Homeomorphisms in Topological Spaces

S.Sakthivel; N.Uma

doi:https://doi.org/10.14445/22315373/IJMTT-V5P507

International Journal of Mathematics Trends and Technology

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Volume 5 | Number 1 | Year 2014 | Article Id. IJMTT-V5P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V5P507

On wgrα - Homeomorphisms in Topological Spaces

S.Sakthivel,N.Uma

Abstract

A bijection f: (X, τ) → (Y, σ) is called weakly generalized regular α- homeomorphism if f and f-1are wgrα-continuous. Also we introduce the new class of maps, namely wgrαc-homeomorphisms which form a subclass of wgrα-homeomorphisms. This class of maps is closed under composition of maps. We prove that the set of all wgrαc-homeomorphisms forms a group under the operation composition of maps.

Keywords

wgrα-closed set, wgrα-open set, wgrα-continuous mapping, wgrα-homeomorphism, wgrαc-homeomorphism.

References

[1]   D. Andrijevic, Semi-preopen Sets, Mat. Vesnik, 38(1986), 24-32.
[2]   I. Arockiarani, Studies On Generalizations of Generalized Closed Sets and Maps in Topological Spaces Ph.D Thesis, Bharathiar University, Coimbatore,(1997).
[3]   K. Balachandran, P. Sundram and H. Maki, On Generalized Continuous Maps In Topological Spaces, Mem. I ac Sci. Kochi Univ. Math., 12(1991), 5-13.
[4]   P. Bhattacharyya and B. K. Lahiri, Semi-generalized Closed Sets in Topology, Indian J. Math. 29(1987) 376-382.
[5]   D.E. Cameron, Properties of S-closed Spaces, Proc. Amer Math. Soc.72 (1978), 581-586.
[6]   S.G. Crossley and S. K. Hildebrand, Semi-topological Properties, Fund.Math., 74(1972),   233-254.
[7]   Y. Gnanambal, Studies On Generalized Pre-regular Closed Sets and Generalization of Locally Closed Sets, Ph.D, Thesis, Bharathiar University, Coimbatore, (1998).
[8]   N. Levine, Semi-open Sets and Semi-continuity in Topological Spaces, Amer. Math. Monthly, 70(1963),36-41.
[9]   N. Levine, Generalized Closed Sets in Topology, Rend. Circ. Mat. Palermo, 19(1970).
[10]   H. Maki, P. Sundaram and K. Balachandran, On Generalized Homeomorphisms in Topological Spaces,   Bull. Fukuoka Univ. Ed, part-III, 40(1991), 13-21.
[11]   S. R. Malghan, Generalized Closed Maps, J. Karnatk Univ. Sci., 27 (1982),82-88.
[12]   A.S. Mashhour, M.E. Abd. EI-Monsef and S.N.EI-Deeb, On Pre- continuous Mappings and Weak Pre-continuous Mappings, Proc. Math. Phy. Soc. Egypt., 53(1982), 47-53.
[13]   A. S. Mashhour, I.A. Hasanein and S. N.EI-Deeb, α-continuous and α-open mappings, Acta. Math. Hung, 41(1983), 213-218.
[14]   N. Nagaveni, Studies On Generalizations of Homeomorphisms in Topological spaces Ph.D., Thesis, Bharathiar University, Coimbatore(1999).
[15]   O. Njastad, On Some Classes of Nearly Open Sets, Paciﬁc J. Math.,15(1965), 961-970.
[16]   A. Pushpalatha, Studies On Generalizations of Mappings in Topological Spaces, Ph.D. Thesis Bharathiar University, Coimbatore(2000).
[17]   M. Sheik John, A Study On Generalizations of Closed Sets On Continuous Maps in Topological and Bitopological Spaces, Ph.D, Thesis Bharathiar University, Coimbatore,(2002).
[18]   M. Sheik John, On w-closed Sets in Topology, Acta Ciencia Indica, 4(2000), 389-392.
[19]   M. K. Singal and A. R. Singal, Almost Continuous Mappings, Yokohaman Math. J. 16(1968),    63-73.
[20]   M. Stone, Application of The Theory of Boolean Rings to General Topology, Trans. Amer. Math. Soc., 41(1937), 374-481.
[21]   P. Sundaram, Studies on Generalizations of Continuous Maps in Topological Spaces, Ph.D, Thesis, Bharathiar University, Coimbatore,(1991).
[22]   A. Vadivel and K. Vairamanickam, rgα-closed Sets and rgα-open Sets in Topological Spaces, Int. Journal of Math. Analysis, Vol. 3, 2009, no. 37, 1803-1819.
[23]   A. Vadivel and K. Vairamanickam, rgα-continuous Maps and Some of Their Properties, Acta Ciencia Indica, (in press).
[24]   A. Vadivel and K. Vairamanickam, rgα-closed Maps and rgα-open Maps in Topological Spaces, Int. Journal of Math. Analysis, Vol. 4, 2010, no. 10, 453-468.
[25]   R.S. Wali, Some Topoics In General and Fuzzy Topological Spaces, Ph.D, Thesis, Karnatak University, Dharwad,(2006).
[26]   A.Jayalakshmi and C. Janaki, On wgrα-closed Sets in Topological spaces, Int. Journal of Math Archieve-3(6),2012, 2386-2392.
[27]   A. Vadivel and K. Vairamanickam, rgα-Homeomorphisms in Topological Spaces, Int. Journal of Math. Analysis, Vol. 4, 2010, no. 18, 881-890.

Citation :

S.Sakthivel,N.Uma, "On wgrα - Homeomorphisms in Topological Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 5, no. 1, pp. 10-15, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V5P507