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The well known natural equivalence [RxS,T] = [R, T s ], valid in the category of sets and set mappings, can be derived in various ways in the category of topological spaces and continuous maps, provided suitable topologies are introduced on the product set R X S and on the set of all continuous maps from <S to T. In this paper we will show how to construct topologies of this kind. The ordinary product topology on R X S and the compact-open topology on T s will be given their proper setting in this context.doi:10.2140/pjm.1970.34.269 fatcat:5ktizr6lenhofiiocdlwrxiow4