Graph topologies on closed multifunctions

Giuseppe Di Maio, Enrico Meccariello, Somashekhar Naimpally

Abstract

In this paper we study function space topologies on closed multifunctions, i.e. closed relations on X x Y using various hypertopologies. The hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of X x Y . We also study several topologies, including one that is derived from the Attouch-Wets filter on the range. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions.


Keywords

Hyperspaces; Function spaces; Graph topologies; Vietoris topology; Fell topology; Uniform convergence on compacta; U-topology; ∆-topology; Proximal ∆-topology; ∆U-topology; Proximal ∆U-topology; Hausdorff-Bourbaki uniformity; ∆-Attouch-Wets filter

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References

A.M. Abd-Allah and R. Brown, A compact-open topology on partial maps with open domains, J. London Math. Soc. (2) 21 (1980), 480-486. http://dx.doi.org/10.1112/jlms/s2-21.3.480

K. Back, Concepts of similarity for utility functions, J. Math. Economics 15 (1986), 129-142. http://dx.doi.org/10.1016/0304-4068(86)90004-2

P. Brandi, R. Ceppitelli and L. Holá, Topological properties of a new graph topology, J. Convex Analysis 5 (1998), 1-12.

G. Beer, Topologies on closed and closed convex sets, Kluwer Publ. (North-Holland 1993).

G. Beer, On the Fell topology, Set-Valued Analysis 1 (1993), 68-80. http://dx.doi.org/10.1007/BF01039292

G. Beer and A. Di Concilio, Uniform continuity on bounded sets and the Attouch-Wets topology, Proc. Amer. Math. Soc. 112 (1991), 235-243. http://dx.doi.org/10.1090/S0002-9939-1991-1033956-1

G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. di Mat. Pura ed Appl. 162 (1992), 367-381. http://dx.doi.org/10.1007/BF01760016

C. Costantini and P. Vitolo, On the in_mum of the Hausdorff metric topologies, Proc. London Math. Soc. 70 (1995), 441-480. http://dx.doi.org/10.1112/plms/s3-70.2.441

I. Del Prete, M. Di Iorio and L. Holá, Graph convergence of set valued maps and its relation to other convergences, Journal of Applied Analysis 6 n. 2 (2000), 213-226. http://dx.doi.org/10.1515/JAA.2000.213

I. Del Prete, M. Di Iorio and L. Holá, Uniform structures on hyperspaces and uniform topologies on spaces of multifunctions, preprint.

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-proximities and hypertopologies, Acta Math. Hungarica 88 (1-2) (2000), 73-93. http://dx.doi.org/10.1023/A:1006752510935

A. Di Concilio and S. Naimpally, Proximal set-open topologies on partial maps, Acta Math. Hungarica 88 (3), (2000), 227-237. http://dx.doi.org/10.1023/A:1006717331197

G. Di Maio and L. Holá, On hit-and-miss topologies, Rend. Acc. Sc. fis. mat. Napoli 57 (1995), 103-124.

G. Di Maio, L. Holá and E. Meccariello, Properties related to first countability and countable compactness in hyperspaces: a new approach, Topology and its Applications, (to appear).

G. Di Maio, E. Meccariello and S. Naimpally, Uniformizing (proximal) ∆-topologies, Topology and its Applications, (to appear).

G. Di Maio and S. Naimpally, Comparison of hypertopologies, Rend. Ist. Mat. Univ. Trieste 22 (1990), 140-161.

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

V.V. Filippov, The topological structure of solution spaces of ordinary differential equations, Russian Math. Surveys 48:1 (1993), 101-154. http://dx.doi.org/10.1070/RM1993v048n01ABEH000986

L. Holá, Topologies on the space of partial maps, Recent Progress in Function Spaces, quaderni di matematica, Vol. 3, Editors: Giuseppe Di Maio and Lubica Holá, (Aracne, 1998), 55-91.

L. Holá and H. Poppe, Fell topology on the space of functions with closed graphs, Rend. Circ. Mat. di Palermo II 48 (1999), 419-430. http://dx.doi.org/10.1007/BF02844333

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

K. Kuratowski, Sur l'espaces des fonctions partielles, Ann. di Mat. Pura ed Appl. (4) 40 (1955), 61-67. http://dx.doi.org/10.1007/BF02416522

R.A. McCoy, Comparison of Hyperspaces and Function Space Topologies, Recent Progress in Function Spaces, quaderni di matematica, Vol. 3, Editors: Giuseppe Di Maio and Lubica Holá, (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-Verlang, (Berlin, 1988).

S. Naimpally, A new uniform convergence for partial functions, Acta Math. Hungarica 88 (1-2) (2000), 45-52. http://dx.doi.org/10.1023/A:1006796325956

S. 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 Holá, (Aracne, 1998), 259-283.

S. Naimpally and C.M. Pareek, Graph topologies for function spaces, II, Ann. Soc. Mathematicae Polonae, Series I: Commentaziones Matematicae XIII (1970), 221-231.

S. Naimpally and B.D. Warrack, Proximity Spaces, Cambridge Tract in Mathematics 59, (Cambridge University Press, 1970).

H. Poppe, Über Graphentopologien für Abbildungsräume I, Bull. Acad. Pol. Sci. Ser. Sci. Math. Aston. Phy. 15 (1967), 71-80.

H. Poppe, Über Graphentopologien für Abbildungsräume II, Math. Nachr. 38 (1968), 89-96. http://dx.doi.org/10.1002/mana.19680380110

G.R. Sell, On the fundamental theory of ordinary differential equations, J. Diff. Eqs. 1 (1965), 370-392. http://dx.doi.org/10.1016/0022-0396(65)90014-8

S.K. Zaremba, Sur certaines familles de courbes en relation avec la theorie des equations differentielles, Rocznik Polskiego Tow. Matemat. 15 (1936), 83-100.

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