Between continuity and set connectedness
DOI:
https://doi.org/10.4995/agt.2010.1727Keywords:
Almost continuous function, D-continuous function, z-continuous function, Quasi θ-continuous function, Faintly continuous function, Functionally Hausdorff space, Zero setAbstract
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.Downloads
References
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, India (1980).
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. Mat. Univ. Bacau 17 (2007), 125–134.
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. Univ. Bacau 14 (2004), 43–56.
J. K. Kohli, D. Singh and R. Kumar, Generalizations of z-supercontinuous functions and D-supercontinuous functions, Appl. Gen. Topol. 9, no. 2 (2008), 239–251.
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.
Downloads
How to Cite
Issue
Section
License
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
