Let (X,d) be a locally compact separable metric space, Y a Fr#xE9;chet space and let C be the of all the closed non-empty susbsets of X. Given #x3C9; #x2208; C let G#x3C9;denote the set of all the graphs of continuous functions in C(#x3C9; Y). Let G=u#x3C9;#x2208;cG#x3C9;We endow G with a new topology called #x3C4;-topology. The topological space (G,#x3C4;) is homeomorphic to the quotient space (C,r)#xD7;C(X,Y)/R with respect to a suitable equivalence relation R. The relationships between #x3C4;-topology and the topologies in C#x3C9;by other Authors are explored. The results here obtained generlize those got in 12 and find applications in the theory of Ordinary and Partial Differential Equations with hereditary structure.
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