Between strong continuity and almost continuity

J.K. Kohli

India

University of Delhi

Department of Mathematics, Hindu College, University of Delhi, Delhi 110 007, India

D. Singh

India

University of Delhi

Department ofMathematics, Sri Aurobindo College, University of Delhi - South Campus, Delhi 110 017, India

Submitted: 2013-09-27

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Accepted: 2013-09-27

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DOI: https://doi.org/10.4995/agt.2010.1726
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Keywords:

(almost) z-supercontinuous function, (almost) Dδ -supercontinuous function, (almost) strongly θ-continuous function, (almost) completely continuous function, nearly paracompact space, almost partition topology

Supporting agencies:

This research was not funded

Abstract:

As embodied in the title of the paper strong and weak variants of continuity that lie strictly between strong continuity of Levine and almost continuity due to Singal and Singal are considered. Basic properties of almost completely continuous functions (≡ R-maps) and δ-continuous functions are studied. Direct and inverse transfer of topological properties under almost completely continuous functions and δ-continuous functions are investigated and their place in the hier- archy of variants of continuity that already exist in the literature is out- lined. The class of almost completely continuous functions lies strictly between the class of completely continuous functions studied by Arya and Gupta (Kyungpook Math. J. 14 (1974), 131-143) and δ-continuous functions defined by Noiri (J. Korean Math. Soc. 16, (1980), 161-166). The class of almost completely continuous functions properly contains each of the classes of (1) completely continuous functions, and (2) al- most perfectly continuous (≡ regular set connected) functions defined by Dontchev, Ganster and Reilly (Indian J. Math. 41 (1999), 139-146) and further studied by Singh (Quaestiones Mathematicae 33(2)(2010), 1–11) which in turn include all δ-perfectly continuous functions initi- ated by Kohli and Singh (Demonstratio Math. 42(1), (2009), 221-231) and so include all perfectly continuous functions introduced by Noiri (Indian J. Pure Appl. Math. 15(3) (1984), 241-250).
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References:

S. P. Arya and R. Gupta, On strongly continuous mappings, Kyungpook Math. J. 14 (1974), 131-143.

D. E. Cameron, Properties of S-closed spaces, Proc. Amer. Math. Soc. 72, no. 3 (1978), 581–586.

D. Carnahan, Some properties related to compactness in topological spaces, Ph.D Thesis, Univ. of Arkansas, 1973.

J. Dontchev, M. Ganster and I. Reilly, More on almost s-continuity, Indian J. Math. 41 (1999), 139-146.

E. K. Van Douwen, Applications of maximal topologies, Topology Appl. 51, no. 2 (1993), 125–139. http://dx.doi.org/10.1016/0166-8641(93)90145-4

J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.

E. Ekici, Generalization of perfectly continuous, regular set-connected and clopen func- tions, Acta. Math. Hungar. 107, no. 3 (2005), 193–206. http://dx.doi.org/10.1007/s10474-005-0190-2

J. K. Kohli, A unified approach to continuous and non-continuous functions II, Bull. Aust. Math. Soc. 41 (1990), 57–74. http://dx.doi.org/10.1017/S0004972700017858

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, Dδ -supercontinuous functions, Indian J. Pure Appl. Math. 34, no. 7 (2003), 1089–1100.

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, δ-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 Mat. 43, no. 3-4 (2010), to appear.

J. K. Kohli, D. Singh and C. P. Arya, Perfectly continuous functions, Stud. Cercet. Stiint. Ser. Mat. Univ. Bacau 18 (2008), 99–110.

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.

N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269. http://dx.doi.org/10.2307/2309695

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

V. J. Mancuso, Almost locally connected spaces, J. Austral. Math. Soc. Ser. A 31 (1981), 421–428. http://dx.doi.org/10.1017/S1446788700024216

A. Mathur, A note on S-closed spaces, Proc. Amer. Math. Soc. 74, no. 2 (1979), 350– 352.

B. M. Munshi and D. S. Bassan, Supercontinuous mappings, Indian J. Pure Appl. Math. 13 (1982), 229-236.

T. Noiri, On δ-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, Strong forms of continuity in topological spaces, Rend. Circ. Mat. Palermo (2) Suppl. 12 (1986), 107–113.

T. Noiri and S. M. Kang, On almost strongly θ-continuous functions, Indian J. Pure Appl. Math. 15, no. 1 (1984), 1–8.

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 Asha Mathur, On nearly compact spaces, Boll. Un. Mat. Ital. (4) 2 (1969), 702–710.

M. K. Singal and S. P. Arya, On nearly paracompact spaces, Mat. Vesnik 6 (21) (1969), 3–16.

M. K. Singal and S. P. Arya, On almost regular spaces, Glasnik Mat. Ser. III 4 (24) (1969), 89–99.

M. K. Singal and S. P. Arya, Almost normal and almost completely regular spaces, Glasnik Mat. Ser. III 5 (25) (1970), 141–152.

M. K. Singal and A. R. Singal, Almost continuous mappings, Yokohama Math. J. 16 (1968), 63–73.

M. K. Singal and A. R. Singal, Mildly normal spaces, Kyungpook Math. J. 13 (1973), 27–31.

D. Singh, D∗-supercontinuous functions, Bull. Cal. Math. Soc. 94, no. 2 (2002), 67–76.

D. Singh, cl-supercontinuous functions, Appl. Gen. Topol. 8, no. 2 (2007), 293–300.

D. Singh, Almost perfectly continuous functions, Quaest. Math. 33 (2010), 1–11. http://dx.doi.org/10.2989/16073606.2010.491187

L. A. Steen and J. A. Seebach, Jr., Counter Examples in Topology, Springer Verlag, New York, 1978.

T. Thompson, S-closed spaces, Proc. Amer. Math. Soc. 60 (1976), 335–338.

N. K. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 2 (78) (1968), 103–118.

A. Wilansky, Between T1 and T2, Amer. Math. Monthly 74 (1967), 261–266. http://dx.doi.org/10.2307/2316017

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