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R-supercontinuous functions

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Języki publikacji
EN
Abstrakty
EN
A new strong variant of continuity called 'R-supercontinuity' is introduced. Basic properties of R-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. It is shown that R-supercontinuity is preserved under the restriction, shrinking and expansion of range, composition of functions, products and the passage to graph function. The class of R-supercontinuous functions properly contains each of the classes of (i) strongly (...)-continuous functions introduced by Noiri and also studied by Long and Herrington; (ii) D-supercontinuous functions; and (iii) F-supercontinuous functions; and so include all z-supercontinuous functions and hence all clopen maps ((...) cl-supercontinuous functions) introduced by Reilly and Vamnamurthy, perfectly continuous functions defined by Noiri and strongly continuous functions due to Levine. Moreover, the notion of r-quotient topology is introduced and its interrelations with the usual quotient topology and other variants of quotient topology in the literature are discussed. Retopologization of the domain of a function satisfying a strong variant of continuity is considered and interrelations among various coarser topologies so obtained are observed.
Wydawca
Rocznik
Strony
703--723
Opis fizyczny
Bibliogr. 36 poz.
Twórcy
autor
autor
autor
  • Department of Mathematics Hindu College University of Delhi, Delhi 110007, India, jk kohli@yahoo.com
Bibliografia
  • [1] P. Alexandroff, Discrete raüme, Mat. Sb. 2 (1937), 501–518.
  • [2] C. E. Aull, Notes on separation by continuous functions, Indag. Math. (N.S.) 31 (1969), 458–461.
  • [3] C. E. Aull, On C-and C_-embeddings, Indag. Math. (N.S.) 37 (1975), 26–33.
  • [4] C. E. Aull, Functionally regular spaces, Indag. Math. (N.S.) (1976), 281–288.
  • [5] N. Bourbaki, General Topology Part I , Hermann, Addison-Wesley (1966).
  • [6] A. J. D’Aristotle, Quasicompact and functionally Hausdorff spaces, J. Austral. Math. Soc. 15 (1973), 319–324.
  • [7] A. S. Davis, Indexed system of neighbourhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886–893.
  • [8] R. F. Dickman Jr., J. R. Porter, θ-closed subsets of Hausdorff spaces, Pacific. J. Math. 59 (1975), 407–415.
  • [9] E. Ekici, Generalizations of perfectly continuous, regular set connected and clopen functions, Acta Math. Hungar. 107(3) (2005), 193–205.
  • [10] G. Gierz, K. H. Hoffman, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, A Compendium of Continuous Lattices, Springer Verlag, Berlin (1980).
  • [11] N. C. Heldermann, Developability and some new regularity axioms, Canad. J. Math. 33(3) (1981), 641–663.
  • [12] E. Hewitt, On two problems of Urysohn, Ann. of Math 47(3) (1946), 503–509.
  • [13] J. K. Kohli, Variants of (complete) regularity and factorizations of (complete) regularity (preprint).
  • [14] J. K. Kohli, R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math. 33(7) (2002), 1097–1108.
  • [15] J. K. Kohli, D. Singh, D-supercontinuous functions, Indian J. Pure Appl. Math. 32(2) (2001), 227–235.
  • [16] J. K. Kohli, D. Singh, Dδ-supercontinuous functions, Indian J. Pure Appl. Math. 34(7) (2003), 1089–1100.
  • [17] J. K. Kohli, D. Singh, Between regularity and complete regularity and a factorization of complete regularity, Stud. Cerc. Mat. Vol. (17) (2007), 125–134.
  • [18] J. K. Kohli, D. Singh, J. Aggarwal, F-supercontinuous functions, Applied General Topology 10(1) (2009), 69–83.
  • [19] J. K. Kohli, D. Singh, J. Aggarwal, On two new factorizations of compactness, preprint.
  • [20] J. K. Kohli, D. Singh, R. Kumar, Some properties of strongly θ-continuous functions, Bull. Calcutta Math. Soc. 100 (2008), 185–196.
  • [21] J. K. Kohli, D. Singh, R. Kumar, J. Aggarwal, Between continuity and set connectedness, Applied General Topology (to appear).
  • [22] N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269.
  • [23] P. E. Long, L. Herrington, Strongly θ-continuous functions, J. Korean Math. Soc. (8) (1981), 21-28.
  • [24] P. E. Long, L. Herrington, The Tθ-topology and faintly continuous functions, Kyungpook Math. J. 22 (1982), 7–14.
  • [25] F. Lorrain, Notes on topological spaces with minimum neighborhoods, Amer. Math. Monthly 76 (1969), 616–627.
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  • [27] B. M. Munshi, D. S. Bassan, Super-continuous mappings, Indian J. Pure Appl. Math. 13 (1982), 229–236.
  • [28] T. Noiri, On δ-continuous functions, J. Korean Math. Soc. 16 (1980), 161–166.
  • [29] T. Noiri, Supercontinuity and some strong forms of continuity, Indian J. Pure. Appl. Math. 15(3) (1984), 241–250.
  • [30] I. L. Reilly, M. K. Vamanamurthy, On super-continuous mappings, Indian J. Pure. Appl. Math. 14(6) (1983), 767–772.
  • [31] N. A. Shanin, On separation in topological spaces, Dokl. Akad. Nauk. SSSR 38 (1943), 110–113.
  • [32] D. Singh, D*-supercontinuous functions, Bull. Calcutta Math. Soc. 94(2) (2002), 67–76.
  • [33] D. Singh, cl-supercontinuous functions, Applied General Topology 8(2) (2007), 293–300.
  • [34] R. M. Stephenson, Spaces for which Stone-Weierstrass theorem holds, Trans. Amer. Math. Soc. 133 (1968), 537–546.
  • [35] L. A. Steen, J. A. Seeback, Jr., Counter Examples in Topology, Springer Verlag, New York, 1978.
  • [36] N. H. Veličko, H-closed topological spaces, Amer. Math Soc. Transl. 78(2) (1968), 103–118.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA4-0032-0027
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