Multivalued function spaces and Atsuji spaces

Somashekhar Naimpally

doi:10.4995/agt.2003.2025

Authors

Somashekhar Naimpally Lakehead University

DOI:

https://doi.org/10.4995/agt.2003.2025

Keywords:

Graph topology, Multi-valued functions, Vietoris topology, Fell topology, Uniform topology, Fell uniform topology, Atsuji spaces, Uniform continuity, Proximal continuity

Abstract

In this paper we present two themes. The first one describes a transparent treatment of some of the recent results in graph topologies on multi-valued functions. The study includes Vietoris topology, Fell topology, Fell uniform topology on compacta and uniform topology on compacta. The second theme concerns when continuity is equivalent to proximal continuity or uniform continuity.

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Author Biography

Somashekhar Naimpally, Lakehead University

Professor Emeritus

References

M. Atsuji Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11-16. http://dx.doi.org/10.2140/pjm.1958.8.11

M. Atsuji, Uniform continuity of continuous functions of uniform spaces, Canadian J. Math. 13 (1961), 657-663. http://dx.doi.org/10.4153/CJM-1961-055-9

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Publ., Holland (1993).

G. Beer, Metric spaces in which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653-658. http://dx.doi.org/10.1090/S0002-9939-1985-0810180-3

G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. 162 (1992), 715-726.

N. Bourbaki, General Topology, Part 2, Addison-Wesley Publishing Company, Reading, MA, (1966).

D. Di Caprio and E. Meccariello, Notes on Separation Axioms in Hyperspaces, Q & A in General Topology 18 (2000), 65-86.

D. Di Caprio and E. Meccariello, G-uniformities LR-uniformities and hypertopologies, Acta Math. Hungarica 88 (2000), 73-93. http://dx.doi.org/10.1023/A:1006752510935

A. Di Concilio and S. Naimpally, Atsuji spaces: continuity versus uniform continuity, Sixth Brazilian Topology Meeting, Campinas, Brazil (1988) (unpublished).

A. Di Concilio, S. Naimpally and P. Sharma, Proximal Hypertopologies, Sixth Brazilian Topology Meeting, Campinas, Brazil (1988) (unpublished).

J. D. Hansard, Function space topologies, Pacific J. Math. 35 (1970), 381-388. http://dx.doi.org/10.2140/pjm.1970.35.381

A. Irudayanathan, Cover-close topologies for function spaces, Gen. Top. And Appl. 10 (1979), 275-282. http://dx.doi.org/10.1016/0016-660X(79)90039-4

J. L. Kelly, General Topology, D. Van Nostrand Company, Princeton, NJ (1960).

K. Kuratowski, Sur l'espaces des fonctions partielles, Ann. Di Mat. Pura ed Appl. 40 (1955), 61-67. http://dx.doi.org/10.1007/BF02416522

R. A. McCoy, Comparison of Hyperspace and Function Space Topologies, Recent Progress in Function Spaces, quaderni di matematica, Vol. 3, Editors: Giuseppe Di Maio and Lubica Hola, Seconda Universita di Napoli, Aracne, (1998), 241-258.

R. A. McCoy, The open-cover topology for function spaces, Fund. Math. 104 (1979), 69-73.

R. A. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lecture Notes in Mathematics # 1315, Springer-Verlag, Berlin (1988).

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

S. A. Naimpally, A brief survey of topologies on function spaces, Recent Progress in Function Spaces, quaderni di matematica, Vol. 3,Editors: Giuseppe Di Maio and Lubica Hola, Seconda Universita di Napoli, Aracne, (1998), 259-283.

S. A. Naimpally and C. M. Pareek, Graph topologies for function spaces II, Ann. Soc. Math. Pol. Series I 13 (1970), 222-231.

S. A. Naimpally and B. D. Warrack, Proximity Spaces, Cambridge Tract # 59 (1970).

H. Poppe, Uber Graphentopologien fur Abbildungsraume I, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phy. 15 (1967), 71-80.