Topologies for ; set-valued functions and their graphs

Author:
Louis J. Billera

Journal:
Trans. Amer. Math. Soc. **155** (1971), 137-147

MSC:
Primary 54.65

MathSciNet review:
0273584

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of topologizing , the set of all closed subsets of a topological space , in such a way as to make continuous functions from a space into precisely those functions with closed graphs. We show there is at most one topology with this property, and if is a regular space, the existence of such a topology implies that is locally compact. We then define the compact-open topology for , which has the desired property for locally compact Hausdorff . The space with this topology is shown to be homeomorphic to a space of continuous functions with the well-known compact-open topology. Finally, some additional properties of this topology are discussed.

**[1]**Richard F. Arens,*A topology for spaces of transformations*, Ann. of Math. (2)**47**(1946), 480–495. MR**0017525****[2]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[3]**Edward G. Effros,*Convergence of closed subsets in a topological space*, Proc. Amer. Math. Soc.**16**(1965), 929–931. MR**0181983**, 10.1090/S0002-9939-1965-0181983-3**[4]**J. M. G. Fell,*A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space*, Proc. Amer. Math. Soc.**13**(1962), 472–476. MR**0139135**, 10.1090/S0002-9939-1962-0139135-6**[5]**Jürgen Flachsmeyer,*Verschiedene Topologisierungen im Raum der abgeschlossenen Mengen*, Math. Nachr.**26**(1963/1964), 321–337 (German). MR**0174026****[6]**Y. Kannai,*Continuity properties of the core of a market*, Econometrica**38**(1970).**[7]**Ernest Michael,*Topologies on spaces of subsets*, Trans. Amer. Math. Soc.**71**(1951), 152–182. MR**0042109**, 10.1090/S0002-9947-1951-0042109-4**[8]**Deane Montgomery and Leo Zippin,*Topological transformation groups*, Interscience Publishers, New York-London, 1955. MR**0073104****[9]**V. I. Ponomarev,*A new space of closed sets and many-valued continuous mappings of bicompacts*, Mat. Sb. (N.S.)**48 (90)**(1959), 191–212 (Russian). MR**0117698**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0273584-0

Keywords:
Set-valued function,
hyperspace,
closed graph,
proper topology,
splitting topology,
conjoining topology,
admissible topology,
multifunction,
space of subsets,
function space,
locally compact space

Article copyright:
© Copyright 1971
American Mathematical Society