On topological groups via a-local functions

[EN] An ideal on a set X is a nonempty collection of subsetsof X which satisfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X; ) an ideal I on X and A ⊂ X, ℜa(A) is defined as ∪{U ∈ a : U − A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by a. A topology, denoted a , finer than a is generated by the basis (I; ) = {V − I : V ∈ a(x); I ∈ I}, and a topology, denoted ⟨ℜa( )⟩ coarser than a is generated by the basis ℜa( ) = {ℜa(U) : U ∈ a}. In this paper A bijection f : (X; ; I) → (X; ;J ) is called a A∗-homeomorphism if f : (X; a ) → (Y; a ) is ahomeomorphism, ℜa-homeomorphism if f : (X;ℜa( )) → (Y;ℜa()) is a homeomorphism. Properties preserved by A∗-homeomorphism are studied as well as necessary and sufficient conditions for a ℜa-homeomorphism to be a A∗-homeomorphism.The authors would like to acknowledge the grant from ministry of high education Malaysia UKMTOPDOWN-ST-06-FRGS0001-2012 for financial support.Al-Omeri, W.; Noorani, MSM.; Al-Omari, A. (2014). On topological groups via a-local functions. Applied General Topology. 15(1):33-42. https://doi.org/10.4995/agt.2014.21263342151W. Al-Omeri, M. Noorani and A. Al-Omari, $a$-local function and its properties in ideal topological space,Fasciculi Mathematici, to appear.Arenas, F. G. (2000). Acta Mathematica Hungarica, 89(1/2), 47-53. doi:10.1023/a:1026773308067Dontchev, J., Ganster, M., & Rose, D. (1999). Ideal resolvability. Topology and its Applications, 93(1), 1-16. doi:10.1016/s0166-8641(97)00257-5E. Ekici, On $a$-open sets, $A^*$-sets and decompositions of continuity and super-continuity, Annales Univ. Sci. Budapest. 51 (2008), 39-51.Erdal, E. (2008). A note on a-open sets and e*-open sets. Filomat, 22(1), 89-96. doi:10.2298/fil0801087eHayashi, E. (1964). Topologies defined by local properties. Mathematische Annalen, 156(3), 205-215. doi:10.1007/bf01363287Hamlett, T. R., & Rose, D. (1992). Remarks on Some Theorems of Banach, McShane, and Pettis. Rocky Mountain Journal of Mathematics, 22(4), 1329-1339. doi:10.1216/rmjm/1181072659T. R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator $Psi$, Bull. U.M.I. 7 4-B (1990), 863-874.Jankovic, D., & Hamlet, T. R. (1990). New Topologies from Old via Ideals. The American Mathematical Monthly, 97(4), 295. doi:10.2307/2324512Khan. (2010). Semi-local Functions in Ideal Topological Spaces. Journal of Advanced Research in Pure Mathematics, 2(1), 36-42. doi:10.5373/jarpm.237.100909Stone, M. H. (1937). Applications of the theory of Boolean rings to general topology. Transactions of the American Mathematical Society, 41(3), 375-375. doi:10.1090/s0002-9947-1937-1501905-7Mukherjee, M. N., Roy, B., & Sen, R. (2007). On extensions of topological spaces in terms of ideals. Topology and its Applications, 154(18), 3167-3172. doi:10.1016/j.topol.2007.08.014Navaneethakrishnan, M., & Paulraj Joseph, J. (2008). g-closed sets in ideal topological spaces. Acta Mathematica Hungarica, 119(4), 365-371. doi:10.1007/s10474-007-7050-1Nasef, A. A., & Mahmoud, R. A. (2002). Some topological applications via fuzzy ideals. Chaos, Solitons & Fractals, 13(4), 825-831. doi:10.1016/s0960-0779(01)00058-3Pettis, B. J. (1951). Remarks on a theorem of E. J. McShane. Proceedings of the American Mathematical Society, 2(1), 166-166. doi:10.1090/s0002-9939-1951-0048012-3R. Vaidyanathaswamy, Set Topology, Chelsea Publishing Company (1960).R. Vaidyanathaswamy, The localization theory in set-topology, Proc. Indian Acad. Sci., 20 (1945), 51-61N.V. Velicko, $H$-Closed Topological Spaces, Amer. Math. Soc. Trans. 78, no. 2 (1968), 103-118