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http://dx.doi.org/10.5831/HMJ.2017.39.4.575

A CLASS OF MAPPINGS BETWEEN Rz-SUPERCONTINUOUS FUNCTIONS AND Rδ-SUPERCONTINUOUS FUNCTIONS  

Prasannan, A.R. (Department of Mathematics, Maharaja Agrasen College, University of Delhi)
Aggarwal, Jeetendra (Department of Mathematics, Shivaji College, University of Delhi)
Das, A.K. (Department of Mathematics, Shri Mata Vaishno Devi University)
Biswas, Jayanta (Department of Mathematics, Delhi University)
Publication Information
Honam Mathematical Journal / v.39, no.4, 2017 , pp. 575-590 More about this Journal
Abstract
A new class of functions called $R_{\theta}$-supercontinuous functions is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity, which already exist in the literature, is elaborated. The class of $R_{\theta}$-supercontinuous functions properly contains the class of $R_z$-supercontinuous functions [39] which in turn properly contains the class of $R_{cl}$-supercontinuous functions [43] and so includes all cl-supercontinuous (clopen continuous) functions ([38], [34]) and is properly contained in the class of $R_{\delta}$-supercontinuous functions [24].
Keywords
${\theta}$-open (closed) set; $r_{\theta}$-open (closed) set; $R_1$-space; ${\theta}$-embeded; $r_{\theta}$-quotient map; $R_{\theta}$-open (closed) map; P-regular space; completely P-regular space; change of topology; P-continuous functions; $P^*$-continuous functions.;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 I.L. Reilly and M.K. Vamanamurthy, On super-continuous mappings, Indian J. Pure. Appl. Math. 14(6) (1983), 767-772.
2 N.A. Shanin, On separation in topological spaces, Dokl. Akad. Nauk. SSSR 38(1943), 110-113.
3 M.K. Singal and S.B. Nimse, z-continuous mappings, Mathematics Student, 66(1-4) (1997), 193-210.
4 D. Singh, D-supercontinuous functions, Bull. Cal. Math. Soc. 94(2) (2002), 67-76.
5 D. Singh, cl-supercontinuous functions, Applied General Topology 8(2) (2007), 293-300.   DOI
6 D. Singh, B.K. Tyagi, J. Aggarwal and J.K. Kohli, Rz-supercontinuous functions, Mathematica Bohemica 140(3) (2014), 329-343.
7 D.W.B. Somerset, Ideal spaces of Banach algebras, Proc. London Math. Soc. 78(3) (1999), 369-400.   DOI
8 T. Soundararajan, Weakly Hausdorff spaces and cardinality of spaces, General Topology and its relations to Modern Analysis and Algebra, Proceedings Kanpur Topology Conference 1968, Academia, Prague, 1971, 301-306.
9 L.A. Steen and J.A. Seeback, Jr., Counter Examples in Topology, Springer Verlag, New York, 1978.
10 B.K. Tyagi, J.K. Kohli and D. Singh, Rcl-supercontinuous functions, Demonstratio Math. 46(1) (2013), 229-244.
11 R. Vaidyanathswamy, Treatise on Set Topology, Chelsea Publishing company, New York, 1960.
12 N.V. Velicko, H-Closed topological spaces, Amer. Math. Soc. Transl. 78(1968), 103-118.
13 C.T. Yang, On paracompact spaces, Proc. Amer. Math. Soc. 5(2) (1954), 185-194.   DOI
14 G.S. Young, Introduction of local connectivity by change of topology, Amer. J. Math. 68(1946), 479-494.   DOI
15 F. Beckhoff, Topologies on the ideal space of a Banach algebra and spectral synthesis, Proc. Amer. Math. Soc. 125(1997), 2859-2866.   DOI
16 S.P. Arya and R. Gupta, On strongly continuous mappings, Kyungpook Math. J. 14(1974), 131-143.
17 F. Beckhoff, Topologies on the spaces of ideals of a Banach algebra, Stud. Math. 115(1995), 189-205.   DOI
18 F. Beckhoff, Topologies of compact families on the ideal space of a Banach algebra, Stud. Math. 118(1996), 63-75.   DOI
19 A.S. Davis, Indexed system of neighbourhoods for general topological spaces, Amer. Math Monthly 68(1961), 886-893.   DOI
20 D. Gauld, M. Mrsevic, I.L. Reilly and M.K. Vamanamurthy, Continuity properties of functions, Coll. Math. Soc. Janos Bolyai 41(1983), 311-322.
21 A.M. Gleason, Universal locally connected refinements, Illinois J. Math. 7(1963), 521-531.
22 E. Hewitt, On two problems of Urysohn, Ann. of Math. 47(3) (1946), 503-509.   DOI
23 J.K. Kohli, A unified approach to continuous and certain non-continuous functions, J. Austral. Math. Soc. Ser. A 48(3) (1990), 347-358.   DOI
24 J.K. Kohli, A class of mappings containing all continuous and all semiconnected mappings, Proc. Amer. Math. Soc. 72(1) (1978), 175-181.   DOI
25 J.K. Kohli, A unified view of (complete) regularity and certain variants of (complete) regularity, Can. J. Math. 36(5) (1984), 783-794.   DOI
26 J.K. Kohli, A framework including the theories of continuous and certain non-continuous functions, Note Mat. 10(1) (1990), 37-45.
27 J.K. Kohli, A unified approach to continuous and certain non-continuous functions II, Bull. Austral. Math. Soc. 41(1) (1990), 57-74.   DOI
28 J.K. Kohli, Change of topology, characterizations and product theorems for semilocally P-spaces, Houston J. Math. 17(3) (1991), 335-350.
29 J.K. Kohli and A.K. Das, New normality axioms and factorizations of normality, Glasnik Mat. 37(57) (2002), 105-114.
30 J.K. Kohli and A.K. Das, On functionally ${\theta}$-normal spaces, Applied General Topology 6(1) (2005), 1-14.   DOI
31 J.K. Kohli and A.K. Das, A class of spaces containing all generalized absolutely closed (almost compact) spaces, Applied General Topology 7(2) (2006), 233-244.   DOI
32 J.K. Kohli, A.K. Das and R. Kumar, Weakly functionally ${\theta}$-normal spaces, ${\theta}$-shrinking of covers and partition of unity, Note di Matematica 19(1999), 293-297.
33 J.K. Kohli and R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math. 33(7) (2002), 1097-1108.
34 J.K. Kohli, D. Singh and J. Aggarwal, R-supercontinuous functions, Demonstratio Mathematica 43(3) (2010), 703-723.
35 J.K. Kohli and D. Singh, $D_{{\delta}}$-supercontinuous functions, Indian J. Pure Appl. Math. 34(7) (2003), 1089-1100.
36 J.K. Kohli and D. Singh, Separation Axioms between regular spaces and $R_{O}$-spaces, Scientific Studies and Research Series Mathematics and Informatics 25(2) (2015), 25-46.
37 J.K. Kohli, D. Singh and J. Aggarwal, F-supercontinuous functions, Applied General Topology 10(1) (2009), 69-83.   DOI
38 J.K. Kohli, D. Singh and J. Aggarwal, R-supercontinuous functions, Demonstratio Mathematica 47(2) (2014), 433-448.
39 J.K. Kohli, D. Singh and C.P. Arya, Perfectly continuous functions, Studii Si Cercetari Stiintifice Ser. Matem. Univ. Bacau Nr. 18(2008), 99-110.
40 J.K. Kohli, D. Singh and R. Kumar, Some properties of strongly ${\theta}$-continuous functions, Bull. Cal. Math. Soc. 100(2008), 185-196.
41 J.K. Kohli, D. Singh and B.K. Tyagi, Quasi perfectly continuous functions and their function spaces, Scientific Studies and Research Series Mathematics and Informatics 21(2) (2011), 23-40.
42 P.E. Long and L. Herrington, Strongly ${\theta}$-continuous functions , J. Korean Math. Soc. 18(1) (1981), 21-28.
43 N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67(1960), 269.   DOI
44 J. Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math. Soc. 148(1970), 265-272.   DOI
45 T. Noiri, Supercontinuity and some strong forms of continuity, Indian J. Pure. Appl. Math. 15(3) (1984), 241-250.
46 B.M. Munshi and D.S. Bassan, Super-continuous mappings, Indian J. Pure Appl. Math.13(1982), 229-236.
47 T. Noiri, On ${\delta}$-continuous functions, J. Korean Math. Soc. 16(2)(1980), 161-166.