Pseudo perfectly continuous functions and closedness/compactness of their function spaces

J.K. Kohli, D. Singh, Jeetendra Aggarwal, Manoj Rana

Abstract

A new class of functions called 'pseudo perfectly continuous' functions is introduced. Their place in the hierarchy of variants of continuity which already exist in the literature is highlighted. The interplay between topological properties and pseudo perfect continuity is investigated. Function spaces of pseudo perfectly continuous functions are considered and sufficient conditions for their closedness and compactness in the topology of pointwise convergence are formulated.

Keywords

(quasi) Perfectly continuous function; Slightly continuous function; Pseudo-partition topology; Alexandroff space

Full Text:

PDF

References

N. Ajmal and J. K. Kohli, Properties of hyperconnected spaces, their mappings into Hausdorff spaces and embeddings into hyperconnected spaces, Acta Math. Hungar. 60, no. 1-2 (1992), 41–49. http://dx.doi.org/10.1007/BF00051755

P. Alexandroff, Discrete raüme, Mat. Sb.2 (1937), 501–518.

A. V. Arhangel’skii, General Topology III, Springer Verlag, Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-662-07413-8

F. Beckhoff, Topologies on the spaces of ideals of a Banach algebra, Stud. Math. 115 (1995), 189–205.

F. Beckhoff, Topologies on the ideal space of a Banach algebra and spectral synthesis, Proc. Amer. Math. Soc. 125 (1997), 2859–2866. http://dx.doi.org/10.1090/S0002-9939-97-03831-8

R.N. Bhaumik, Role of regular G_-subsets in set-theoretic topology, The Mathematics Student 70, no. 1-4 (2001), 99–104.

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

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

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

S. Fomin, Extensions of topological spaces, Ann. of Math. 44 (1943), 471–480. http://dx.doi.org/10.2307/1968976

A. M. Gleason, Universal locally connected refinements, Illinois J. Math. 7 (1963), 521–531.

N. C. Heldermann, Developability and some new regularity axioms, Canadian J. Math. 33, no. 3 (1981), 641–663. http://dx.doi.org/10.4153/CJM-1981-051-9

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. L. Kelley, General Topology, Van Nostrand, New York, 1955.

J. K. Kohli, A class of mappings containing all continuous and all semiconnected mappings , Proc. Amer. Math. Soc. 72, no. 1 (1978), 175–181. http://dx.doi.org/10.1090/S0002-9939-1978-0493941-9

J. K. Kohli, A class of spaces containing all connected and all locally connected spaces, Math. Nachricten 82 (1978), 121–129. http://dx.doi.org/10.1002/mana.19780820113

J. K. Kohli, Change of topology, characterizations and product theorems for semilocally P-spaces, Houston J. Math. 17, no. 3 (1991), 335–350.

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 J. Aggarwal, Quasi cl-supercontinuous functions and their function spaces, Demonstratio Math. 45, no. 3 (2012), 677–697.

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. 34, no. 7 (2003), 1089–1100.

J. K. Kohli and D. Singh, Between weak continuity and set connectedness, Studii Si Cercetari Stiintifice Ser. Matem. Univ. Bacau Nr. 15 (2005), 55–65.

J. K. Kohli and D. Singh, Between compactness and quasicompactness, Acta Math. Hungar. 106, no. 4 (2005), 317–329. http://dx.doi.org/10.1007/s10474-005-0022-4

J. K. Kohli and D. Singh, Between regularity and complete regularity and a factorization of complete regularity, Studii Si Cercetari Seria Matematica 17 (2007), 125–134.

J. K. Kohli and D. Singh, Function spaces and strong variants of continuity, Applied General Topology 9, no. 1 (2008), 33–38.

J. K. Kohli and D. Singh,Almost cl-supercontinuous functions, Applied General Topology 10, no. 1 (2009), 1–12.

J. K. Kohli and D. Singh, _-perfectly continuous functions, Demonstratio Mathematica 42, no. 1 (2009), 221–231.

J. K. Kohli and D. Singh, On certain generalizations of supercontinuity / _-continuity, Scientific Studies and Research Series Mathematics and Informatics (to appear).

J. K. Kohli and D. Singh, Pseudo strongly _-continuous functions, preprint.

J. K. Kohli, D. Singh and J. Aggarwal, R-supercontinuous functions, Demonstratio Mathematica 43, no. 3 (2010), 703–723.

J. K. Kohli, D. Singh and J. Aggarwal, Pseudo cl-supercontinuous functions and closedness / compactness of their function spaces, preprint.

J. K. Kohli, D. Singh and C. P. Arya, Perfectly continuous functions, Stud. Cerc. St. Ser. Mat. Nr. 18 (2008), 99–110.

J. K. Kohli, D. Singh and R. Kumar, Quasi z-supercontinuous functions and pseudo z-supercontinuous functions, Studii Si Cercetari Stiintifice Ser. Matem. Univ. Bacau Nr. 14 (2004), 43–56.

J. K. Kohli, D. Singh and R. Kumar, Generalizations of z-supercontinuous functions and D_-supercontinuous functions, Applied General Topology 9, no. 2 (2008), 239–251.

J. K. Kohli, D. Singh, R. Kumar and J. Aggarwal, Between continuity and set connectedness , Applied General Topology 11, no. 1 (2010), 43–55.

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, no. 2 (2011), 23–40.

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 space, Amer. Math. Monthly, 68 (1961), 44–46. http://dx.doi.org/10.2307/2311363

P. E. Long and L. Herrington, Strongly _-continuous functions, J. Korean Math. Soc. 8 (1981), 21–28.

P. E. Long and L. Herrington, T_-topology and faintly continuous functions, Kyungpook Math. J. 22 (1982), 7–14.

F. Lorrain, Notes on topological spaces with minimum neighbourhoods, Amer. Math. Monthly 76 (1969), 616–627. http://dx.doi.org/10.2307/2316662

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. 31 (1981), 421–428. http://dx.doi.org/10.1017/S1446788700024216

A. K. Misra, A topological view of P-spaces, General Topology and its Applications 2 (1972), 349–362. http://dx.doi.org/10.1016/0016-660X(72)90026-8

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

S. A. Naimpally, On strongly continuous functions, Amer. Math. Monthly 74 (1967), 166–168. http://dx.doi.org/10.2307/2315609

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

T. Noiri and V. Popa, Weak forms of faint continuity, Bull. Math. de la Soc. Sci. Math. de la Roumanie 34 (82) (1990), 263–270.

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. B. Nimse, z-continuous mappings, The Mathematics Student 66, no. 1-4 (1997), 193–210.

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.

D. Singh, Almost perfectly continuous functions, Quaestiones Mathematicae 33, no. 2 (2010), 211–221. http://dx.doi.org/10.2989/16073606.2010.491187

D. W. B. Somerset, Ideal spaces of Banach algebras, Proc. London Math. Soc. 78, no. 3 (1999), 369–400. http://dx.doi.org/10.1112/S0024611599001677

R. Staum, The Algebra of bounded continuous functions into a nonarchimedean field, Pac. J. Math. 50, no. 1 (1974), 169–185. http://dx.doi.org/10.2140/pjm.1974.50.169

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. K. Veliˇcko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968), 103–118.

G. S. Young, Introduction of local connectivity by change of topology, Amer. J. Math. 68 (1946), 479–494. http://dx.doi.org/10.2307/2371828

Abstract Views

1141
Metrics Loading ...

Metrics powered by PLOS ALM




Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt