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Function Space Topologies for Connectivity and Semi-Connectivity Functions

Published online by Cambridge University Press:  20 November 2018

Somashekhar Amrith Naimpally
Somashekhar Amrith Naimpally*
Affiliation:
University of Alberta, Edmonton
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Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.

Type
Notes and Problems
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 3 , August 1966 , pp. 349 - 352
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hocking, J.G. and Young, G. S., Topology. Addison-Wesley, Reading, Mass. 1961.Google Scholar
2. Kelley, J. L., General Topology, Van No strand Princeton, N.J., 1955.Google Scholar
3. Naimpally, S.A., Graph topology for function spaces, Notices of the Amer. Math. Soc. 79, (1965) p. 74.Google Scholar
4. Sierpinski, W., Sur une propriete de fonctions reelles quelconques definie dans les espaces metriques, Le Matematicae (Catenia) 8 (1953), 73-78.Google Scholar