On topological groups via a-local functions

• Al-Omeri, Wadei ^[1] ; Noorani, M. Salmi Md. ^[1] ; Al-Omari, A. ^[2]
  1. [1] National University of Malaysia

     National University of Malaysia

     Malasia

     
  2. [2] Al al-Bayt University

     Al al-Bayt University

     Jordania

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 15, Nº. 1, 2014, págs. 33-42
• Idioma: inglés
• DOI: 10.4995/agt.2014.2126
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 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...
  • 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.