Applied General Topology
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On topological groups via a-local functions
Abstract
of 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 a
homeomorphism, 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.
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References
W. Al-Omeri, M. Noorani and A. Al-Omari, $a$-local function and its properties in ideal topological space,Fasciculi Mathematici, to appear.
W. AL-Omeri, M. Noorani and A. AL-Omari, On $Re_ a$- operator in ideal topological spaces, submitted.
F. G. Arenas, J. Dontchev and M. L. Puertas, Idealization of some weak separation axioms, Acta Math. Hungar. 89, no. 1-2 (2000), 47-53.
(http://dx.doi.org/10.1023/A:1026773308067)
J. Dontchev, M. Ganster, D. Rose, Ideal resolvability, Topology Appl. 93 (1999), 1-16.
(http://dx.doi.org/10.1016/S0166-8641(97)00257-5)
E. Ekici, On $a$-open sets, $A^*$-sets and decompositions of continuity and super-continuity, Annales Univ. Sci. Budapest. 51 (2008), 39-51.
E. Ekici, A note on $a$-open sets and $e^*$-open sets, Filomat 22, no. 1 (2008), 89-96.
(http://dx.doi.org/10.2298/FIL0801087E)
E. Hayashi, Topologies denfined by local properties, Math. Ann. 156 (1964), 205-215.
(http://dx.doi.org/10.1007/BF01363287)
T. R. Hamlett and D. Rose, Remarks on some theorems of Banach, McShane, and Pettis, Rocky Mountain J. Math. 22, no. 4 (1992), 1329-1339.
(http://dx.doi.org/10.1216/rmjm/1181072659)
T. R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator $Psi$, Bull. U.M.I. 7 4-B (1990), 863-874.
D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310.
(http://dx.doi.org/10.2307/2324512)
K. Kuratowski, Topology, Vol. I. NewYork: Academic Press (1966).
M. Khan and T. Noiri, Semi-local functions in ideal topological spaces. J. Adv. Res. Pure Math. 2, no. 1 (2010), 36-42.
(http://dx.doi.org/10.5373/jarpm.237.100909)
M. H. Stone, Application of the Theory of Boolean Rings to General Topology, Trans. Amer. Math. Soc. 41 (1937), 375-481.
(http://dx.doi.org/10.1090/S0002-9947-1937-1501905-7)
M. N. Mukherjee, R. Bishwambhar and R. Sen, On extension of topological spaces in terms of ideals, Topology Appl. 154 (2007), 3167-3172.
(http://dx.doi.org/10.1016/j.topol.2007.08.014)
M. Navaneethakrishnan and J. Paulraj Joseph, $g$-closed sets in ideal topological spaces, Acta Math. Hungar. 119, no. 4 (2008), 365-371.
(http://dx.doi.org/10.1007/s10474-007-7050-1)
A. A. Nasef and R. A. Mahmoud, Some applications via fuzzy ideals, Chaos, Solitons and Fractals 13 (2002), 825-831.
(http://dx.doi.org/10.1016/S0960-0779(01)00058-3)
R. L. Newcomb, Topologies which are compact modulo an ideal, Ph. D. Dissertation, Univ. of Cal. at Santa Barbara. (1967). J.C. Oxtoby, measure and catagory, springer verlag, New York (1986).
B. J. Pettis, Remark on a theorem of E. J. McShane, Proc. Amer. Math. Soc. 2 (1951), 166-171.
(http://dx.doi.org/10.1090/S0002-9939-1951-0048012-3)
R. Vaidyanathaswamy, Set Topology, Chelsea Publishing Company (1960).
R. Vaidyanathaswamy, The localization theory in set-topology, Proc. Indian Acad. Sci., 20 (1945), 51-61
N.V. Velicko, $H$-Closed Topological Spaces, Amer. Math. Soc. Trans. 78, no. 2 (1968), 103-118.
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