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

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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. 
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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     title = {Function {Space} {Topologies} for {Connectivity} and {Semi-Connectivity} {Functions}},
     journal = {Canadian mathematical bulletin},
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     year = {1966},
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     doi = {10.4153/CMB-1966-044-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-044-4/}
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