Geodesic
Function Space Topologies for Connectivity and Semi-Connectivity Functions
Canadian mathematical bulletin, Tome 9 (1966) no. 3, pp. 349-352

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Naimpally, Somashekhar Amrith. Function Space Topologies for Connectivity and Semi-Connectivity Functions. Canadian mathematical bulletin, Tome 9 (1966) no. 3, pp. 349-352. doi: 10.4153/CMB-1966-044-4
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[1] 1. Hocking, J.G. and Young, G. S., Topology. Addison-Wesley, Reading, Mass. 1961. Google Scholar

[2] 2. Kelley, J. L., General Topology, Van No strand Princeton, N.J., 1955. Google Scholar

[3] 3. Naimpally, S.A., Graph topology for function spaces, Notices of the Amer. Math. Soc. 79, (1965) p. 74. Google Scholar

[4] 4. Sierpinski, W., Sur une propriete de fonctions reelles quelconques definie dans les espaces metriques, Le Matematicae (Catenia) 8 (1953), 73-78. Google Scholar

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