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An ascoli theorem for multi-valued functions

Published online by Cambridge University Press:  09 April 2009

Vincent J. Mancuso
Vincent J. Mancuso
Affiliation:
St. John's University Jamaica, New York, U.S.A.
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The concept of simultaneous or collective continuity of a family of single valued functions was introduced by Gale [3] for regular spaces to replace equicontinuiry in metric spaces. Smithson [6] extended the standard point-open and compact-open function space topologies to include multi-valued functions. The aim of this paper is to use these topologies and extend the notion of collective continuity in order to obtain an Ascoli type theorem for multi-valued functions analogous to Theorem 1 in [3, p. 304]. We have the following theorem in mind:

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 466 - 472
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Berge, C., Topological Spaces (New York, 1963).Google Scholar
[2]Day, J. M. and Franklin, S. P., ‘Spaces of continuous relations’, Math. Annalen 167 (1967), 289–292CrossRefGoogle Scholar
[3]Gale, D., ‘Compact sets of functions and function rings’, Proc. Amer. Math. Soc. 43 (1950), 303–308.CrossRefGoogle Scholar
[4]Kelley, J. L., General Topology (Princeton, 1955).Google Scholar
[5]Ponomarev, V. I., ‘Properties of topological spaces preserved under multi-valued continuous mappings’, Amer. Math. Soc. Transl.Google Scholar
[6]Smithson, R. E., ‘Topologies on sets of relations’, to appear.Google Scholar
[7]Smithson, R. E., ‘Some general properties of multi-valued functions’, Pac. J. Math. 15 (1965), 681–703.CrossRefGoogle Scholar
[8]Strother, W., ‘Fixed points, fixed sets, and M-retracts’, Duke J. Math. 22 (1955), 551–556.CrossRefGoogle Scholar