R-spaces and closedness/completeness of certain function spaces in the topology of uniform convergence

D. Singh; J.K. Kohli

doi:10.4995/agt.2014.3029

Authors

D. Singh University of Delhi
J.K. Kohli University of Delhi

DOI:

https://doi.org/10.4995/agt.2014.3029

Keywords:

R space, ultra Hausdorff space, initial property, monoreflective (epireflective) subcategory, R_cl-supercontinuous function, topology of uniform convergence

Abstract

It is shown that the notion of an âˆ’ cl R space (Demonstratio Math. 46(1) (2013), 229-244) fits well as a separation axiom between zero dimensionality and âˆ’ 0 R spaces. Basic properties of âˆ’ cl R spaces are studied and their place in the hierarchy of separation axioms that already exist in the literature is elaborated. The category of âˆ’ cl R spaces and continuous maps constitutes a full isomorphism closed, monoreflective (epireflective) subcategory of TOP. The function space cl R (X, Y) of all âˆ’ cl R supercontinuous functions from a space X into a uniform space Y is shown to be closed in the topology of uniform convergence. This strengthens and extends certain results in the literature (Demonstratio Math. 45(4) (2012), 947-952).

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Author Biography

D. Singh, University of Delhi

Deptt. of Mathematics

Asstt. Professor

References

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

S. P. Arya and M. Deb, On mapping almost continuous in the sense of Frol'ık, Math. Student 41 (1973), 311-321.

C. E. Aull, Functionally regular spaces, Indag. Math. 38 (1976), 281-288. https://doi.org/10.1016/1385-7258(76)90066-4

A. Császár, General Topology, Adam Higler Ltd., Bristol, 1978.

A. S. Davis, Indexed system of neighbourhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886-893. https://doi.org/10.2307/2311686

Z. Frolík, Remarks concerning the invariance of Baire spaces under mapping, Czechoslovak Math. J. 11, no. 3 (1961), 381-385. https://doi.org/10.21136/CMJ.1961.100466

M. Ganster, On strongly s-regular spaces, Glasnik Mat. 25, no. 45 (1990), 195-201.

K. R. Gentry, and H. B. Hoyle, III, Somewhat continuous functions, Czechoslovak Math. J. 21, no. 1 (1971), 5-12. https://doi.org/10.21136/CMJ.1971.100999

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

H. B. Hoyle, III, Function spaces for somewhat continuous functions, Czechoslovak Math. J. 21, no. 1 (1971), 31-34. https://doi.org/10.21136/CMJ.1971.101001

H. Herrlich and G. E. Strecker, Category Theory An Introduction, Allyn and Bacon Inc. Bostan, 1973.

J. L. Kelly, General Topology, Van Nostrand, New York, 1955.

S. Kempisty, Sur les functions quasicontinuous, Fund. Math. 19 (1932), 184-197. https://doi.org/10.4064/fm-19-1-184-197

J. K. Kohli and J. Aggarwal, Closedness of certain classes of functions in the topology of uniform convergence, Demonstratio Math. 45, no. 4 (2012), 947-952. https://doi.org/10.1515/dema-2013-0413

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, no. 7 (2003), 1089-1100.

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, Separation axioms between regular spaces and R0 spaces, preprint.

J. K. Kohli and D.Singh, Separation axioms between functionally regular spaces and R0 spaces, preprint.

J. K. Kohli, B. K. Tyagi, D. Singh and J. Aggarwal, R-supercontinuous functions, Demonstratio Math. 47, no. 2 (2014), 433-448. https://doi.org/10.2478/dema-2014-0034

N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 34-41. https://doi.org/10.2307/2312781

J.Mack, Countable paracompactness and weak normality properties, Trans. Amer.Math. Soc. 148 (1970), 265-272. https://doi.org/10.1090/S0002-9947-1970-0259856-3

A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, -continuous and -open mappings, Acta Math. Hungar. 41 1983, 213-218. https://doi.org/10.1007/BF01961309

S. A. Naimpally, Function space topologies for connectivity and semiconnectivity functions , Canad. Math. Bull. 9 (1966), 349-352. https://doi.org/10.4153/CMB-1966-044-4

S. A. Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267-272. https://doi.org/10.1090/S0002-9947-1966-0192466-4

O. Njástad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961-970. https://doi.org/10.2140/pjm.1965.15.961

N. A. Shanin, On separation in topological spaces, Dokl. Akad. Nauk SSSR, 38 (1943), 110-113.

W. Sierpinski, Sur une propriété de functions réelles quelconques, Matematiche (Catania) 8 (1953), 43-48.

M. K. Singal and S. B. Niemse, z-continuous mappings, The Mathematics Student 66, no. 1-4 (1997), 193-210.

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. https://doi.org/10.4995/agt.2007.1899

D. Singh, B. K. Tyagi, J. Aggarwal and J. K. Kohli, Rz-supercontinuous functions, Math. Bohemica, to appear.

J. R. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249-263. https://doi.org/10.4064/fm-47-3-249-263

R. Staum, The Algebra of bounded continuous functions into a nonarchimedean field, Pac. J. Math. 50, no. 1 (1974), 169-185. https://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. https://doi.org/10.1007/978-1-4612-6290-9

B. K. Tyagi, J. K. Kohli and D. Singh, Rcl-supercontinuous functions, Demonstratio Math. 46, no. 1 (2013), 229-244. https://doi.org/10.1515/dema-2013-0437

R. Vaidyanathswamy, Treatise on Set Topology, Chelsa Publishing Company, New York, 1960.

W. T. Van East and H. Freudenthal, Trennung durch stetige Functionen in topologishen Raümen, Indag. Math. 15 (1951), 359-368. https://doi.org/10.1016/S1385-7258(51)50051-3

N. K. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968), 103-118. https://doi.org/10.1090/trans2/078/05

G. J. Wong, On S-closed spaces, Acta Math. Sinica, 24 (1981), 55-63.

C. T. Yang, On paracompact spaces, Proc. Amer. Math. Soc. 5, no. 2 (1954), 185-194. https://doi.org/10.1090/S0002-9939-1954-0062418-0