Between continuity and set connectedness

• Kohli, J.K. ^[1] ; Singh, D. ^[1] ; Kumar, Rajesh ^[1] ; Aggarwal, Jeetendra ^[1]
  1. [1] University of Delhi

     University of Delhi

     India

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 11, Nº. 1, 2010, págs. 43-55
• Idioma: inglés
• DOI: 10.4995/agt.2010.1727
• Enlaces
  • Texto completo
• Resumen

  • Two new weak variants of continuity called 'R-continuity'and 'F-continuity' are introduced. Their basic properties are studied and their place in the hierarchy of weak variants of continuity, that already exist in the literature, is elaborated. The class of R-continuous functions properly contains the class of continuous functions and is strictly contained in each of the three classes of (1) faintly continu-ous functions studied by Long and Herrignton (Kyungpook Math. J.22(1982), 7-14); (2) D-continuous functions introduced by Kohli (Bull.Cal. Math. Soc. 84 (1992), 39-46), and (3) F-continuous functions which in turn are strictly contained in the class of z-continuous functions studied by Singal and Niemse (Math. Student 66 (1997), 193-210).So the class of R-continuous functions is also properly contained in each of the classes of D∗-continuous functions, D-continuous function and set connected functions.

• Referencias bibliográficas
  • S. P. Arya and M. Deb, On -continuous mappings, Math. Student 42 (1974), 81–89.
  • C. E. Aull, Notes on separation by continuous functions, Indag. Math. 31 (1969), 458-461.
  • C. E. Aull, Functionally regular spaces, Indag. Math. 38 (1976), 281–288.
  • A. S. Davis, Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886–893. http://dx.doi.org/10.2307/2311686
  • K. K. Dube, A note on R0-topological spaces, Mat. Vesnik 11, no. 26 (1974), 203–208.
  • J. Dugundji, Topology, Allyn and Bacon, Boston 1966.
  • S. Fomin, Extensions of topological spaces, Ann. of Math. 44 (1943), 471–480. http://dx.doi.org/10.2307/1968976
  • N. C. Heldermann, Developability and some new regularity axioms, Canad. J. Math. 33, no. 3 (1981), 641–663. http://dx.doi.org/10.4153/CJM-1981-051-9
  • E. Hewitt, On two problems of Urysohn, Ann. of Math. 47, no. 3 (1946), 503–509. http://dx.doi.org/10.2307/1969089
  • R. C. Jain, The role of regularly open sets in general topology, Ph. D. Thesis, Meerut University, Institute of Advanced Studies, Meerut,...
  • J. K. Kohli, D-continuous functions, D-regular spaces and D-Hausdorff spaces, Bull. Cal. Math. Soc. 84 (1992), 39–46.
  • J. K. Kohli and R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math. 33, no. 7 (2002), 1097–1108.
  • J. K. Kohli and D. Singh, D-supercontinuous functions, Indian J. Pure Appl. Math. 32, no. 2 (2001), 227–235.
  • J.K. Kohli and D. Singh, D-supercontinuous functions, Indian J. Pure Appl. Math. 34 (7)(2003), 1089-1100.
  • J. K. Kohli and D. Singh, Between weak continuity and set connectedness, Stud. Cercet. Stiint. Ser. Mat. Univ. Bacau 15 (2005), 55–65.
  • J. K. Kohli and D. Singh, Between regularity and complete regularity and a factorization of complete regularity, Stud. Cercet. Stiint. Ser....
  • J. K. Kohli and D. Singh, Almost cl-supercontinuous functions, Appl. Gen. Topol. 10, no. 1 (2009), 1–12.
  • J. K. Kohli and D. Singh, perfectly continuous functions, Demonstratio Math. 42, no. 1 (2009), 221–231.
  • J. K. Kohli, D. Singh and J. Aggarwal, F-supercontinuous functions, Appl. Gen. Topol. 10, no. 1 (2009), 69–83
  • J. K. Kohli, D. Singh and J. Aggarwal, R-supercontinuous functions, Demonstratio Math. 43 (2010), to appear.
  • J. K. Kohli, D. Singh and R. Kumar, Quasi z-supercontinuous functions and pseudo z-supercontinuous functions, Stud. Cercet. Stiint. Ser. Mat....
  • J. K. Kohli, D. Singh and R. Kumar, Generalizations of z-supercontinuous functions and D-supercontinuous functions, Appl. Gen. Topol. 9, no....
  • J. H. Kwak, Set connected mapping, Kyungpook Math. J. 11 (1971), 169–172.
  • N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269. http://dx.doi.org/10.2307/2309695
  • N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly 68 (1961), 44–46. http://dx.doi.org/10.2307/2311363
  • P. E. Long and L. L. Herrington, Functions with strongly closed graphs, Boll. Unione Mat. Ital. 12 (1975), 381–384.
  • P. E. Long and L. L. Herrington, Strongly -continuous functions, J. Korean Math. Soc. 18 (1981).
  • P. E. Long and L. L. Herrington, The T -topology and faintly continuous functions, Kyungpook Math. J. 22 (1982), 7–14.
  • J.Mack, Countable paracompactness and weak normality properties, Trans. Amer.Math. Soc. 148 (1970), 265–272. http://dx.doi.org/10.1090/S0002-9947-1970-0259856-3
  • B. M. Munshi and D. S. Bassan, Supercontinuous mappings, Indian J. Pure Appl. Math. 13 (1982), 229–236.
  • T. Noiri, On -continuous functions, J. Korean Math. Soc. 16 (1980), 161–166.
  • T. Noiri. Supercontinuity and some strong forms of continuity, Indian J. Pure Appl. Math. 15, no. 3 (1984), 241–250.
  • T. Noiri and V. Popa, Weak forms of faint continuity, Bull. Math. de la Soc. Sci. Math. de la Roumanic 34, no. 82 (1990), 263–270.
  • I. L. Reilly and M. K. Vamanamurthy, On supercontinuous mappings, Indian J. Pure Appl. Math. 14 (1983), 767–772.
  • M. K. Singal and S. B. Niemse, z-continuous mappings, Math. Student 66, no. 1-4 (1997), 193–210.
  • M. K. Singal and A. R. Singal, Almost continuous mappings, Yokohama Math. Jour. 16 (1968), 67–73.
  • D. Singh, D∗-continuous functions, Bull. Cal. Math. Soc. 91, no. 5 (1999), 385–390.
  • D. Singh, cl-supercontinuous functions, Appl. Gen. Topol. 8, no. 2 (2007), 293–300.
  • L. A. Steen and J. A. Seebach, Jr., Counter Examples in Topology, Springer Verlag, New York, 1978. http://dx.doi.org/10.1007/978-1-4612-6290-9
  • N. V. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968),103–118.