Generalizations of Z-supercontinuous functions and Dδ-supercontinuous functions
Keywords:(almost) z-supercontinuous function, (almost) Dδ-supercontinuous function, (almost) strongly θ-continuous function, Almost continuous function, δ-continuous function, faintly continuous function, uθ-closed graph, θ-closed graph, uθ-limit point, θ-limit po
Two new classes of functions, called ‘almost z-supercontinuous functions’ and ’almost Dδ-supercontinuous functions’ are introduced. The class of almost z-supercontinuous functions properly includes the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33(7), (2002), 1097-1108) as well as the class of almost clopen maps due to Ekici (Acta. Math. Hungar. 107(3), (2005), 193-206) and is properly contained in the class of almost Dδ-supercontinuous functions which in turn constitutes a proper subclass of the class of almost strongly θ-continuous functions due to Noiri and Kang (Indian J. Pure Appl. Math. 15(1), (1984), 1-8) and which in its turn include all δ-continuous functions of Noiri (J. Korean Math. Soc. 16 (1980), 161-166). Characterizations and basic properties of almost z-supercontinuous functions and almost Dδ-supercontinuous functions are discussed and their place in the hierarchy of variants of continuity is elaborated. Moreover, properties of almost strongly θ-continuous functions are investigated and sufficient conditions for almost strongly θ-continuous functions to have u θ-closed (θ-closed) graph are formulated.
D. Carnahan, Locally nearly compact spaces, Boll. Mat. Un. Ital. 6, no. 4 (1972), 146–153.
A. K. Das, A note on θ-Hausdorff spaces, Bull. Cal. Math. Soc. 97, no. 1(2005), 15–20.
J. Dontchev, M. Ganster and I. Reilly, More on almost s-continuity, Indian J. Math. 41 (1999), 139–146.
E. Ekici, Generalization of perfectly continuous, regular set-connected and clopen functions, Acta. Math. Hungar. 107, no. 3 (2005), 193–206.
Z-Frolik, Generalizations of compact and Lindelöf spaces, Czechoslovak Math J. 13, no. 84 (1959), 172–217 (Russian) MR 21 # 3821.
N. C. Heldermann, Developability and some new regularity axioms, Can. J. Math. 33, no. 3(1981), 641–668.
E. Hewitt, On two problems of Urysohn, Ann. of Math. 47, no.3 (1946), 503–509. http://dx.doi.org/10.2307/1969089
J. K. Kohli and A. K. Das, New normality axioms and decompositions of normality, Glasnik Mat. 37, no. 57 (2002), 105–114.
J. K. Kohli and A. K. Das, On functionally θ-normal spaces, Applied General Topology 6, no. 1 (2005), 1–14.
J. K. Kohli and A. K. Das, A class of spaces containing all generalized absolutely closed (almost compact) spaces, Applied General Topology 7, no. 2 (2006), 233–244.
J. K. Kohli, A. K. Das and R. Kumar, Weakly functionally θ-normal spaces, θ-shrinking of covers and partition of unity, Note di Matematica 19 (1999), 293–297.
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, Dδ-supercontinuous functions, Indian J.Pure Appl. Math. 34, no. 7 (2003), 1089–1100.
J. K. Kohli and D. Singh, Between compactness and quasicompactness, Acta Math. Hungar. 106, no. 4 (2005), 317–329.
J. K. Kohli and D. Singh, Between weak continuity and set connectedness, Studii Si Cercetari Stintifice Seria Mathematica 15 (2005), 55–65.
J. K. Kohli and D. Singh, Between regularity and complete regularity and a factorization of complete regularity, Studii Si Cercetari Stintifice Seria Mathematica, 17 (2007), 125–134.
J. K. Kohli and D. Singh, Almost cl-supercontinuous functions, Applied General Topology, to appear.
N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269.
P. E. Long and L. Herrington, Strongly θ-continuous functions, J. Korean Math. Soc. 8 (1981), 21–28.
P. E. Long and L. Herrington, The Tθ-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.
B. M. Munshi and D. S. Bassan, Supercontinuous mappings, Indian J. Pure Appl. Math. 13 (1982), 229–236.
T. Noiri, On functions with strongly closed graph, Acta Math. Hungar. 32 (1978), 1–4.
T. Noiri, On δ-continuous functions, J. Korean Math. Soc. 18 (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, Strong forms of continuity in topological spaces, Suppl. Rendiconti Circ. Mat. Palermo, II 12 (1986), 107–113.
T. Noiri and S. M. Kang, On almost strongly -continuous functions, Indian J. Pure Appl. Math. 15, no. 1 (1984), 1–8.
J. R. Porter and J. Thomas, On H-closed spaces and minimal Hausdorff spaces, Trans. Amer. Math. Soc. 138 (1969), 159–170.
I. L. Reilly and M. K. Vamanamurthy, On super-continuous mappings, Indian J. Pure. Appl. Math. 14, no. 6 (1983), 767–772.
M. K. Singal and S. P. Arya, On almost regular spaces, Glasnik Mat. 4, no. 24 (1969), 89–99.
M. K. Singal and S. P. Arya, On almost normal and almost completely regular spaces, Glasnik Mat. 5, no. 25 (1970), 141–152.
M. K. Singal and A. Mathur, On nearly compact spaces, Boll. Un. Mat. Ital. 2, no. 4 (1969), 702–710.
M. K. Singal and A. R. Singal, Almost continuous mappings, Yokohama Math. J. 16 (1968), 63–73.
D. Singh, cl-supercontinuous functions, Applied General Topology 8, no. 2 (2007), 293–300.
S. Sinharoy and S. Bandyopadhyay, On θ-completely regular spaces and locally θâˆ’ H-closed spaces, Bull. Cal. Math. Soc. 87 (1995), 19–28.
N. K. Veliˇcko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968), 103–118.
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