For the category of topological spaces there is an adjunction between the coproduct functor, and the Hom functor (−) X ⊣ (−)X . The left side of this adjunction only exists in categories which admit arbitrary coproducts, but the Hom(X,−) = (−)X functor always exists. We attempt to explain this discrepancy by searching for categories which have an enrichment of the Hom functor with a left adjoint F(−,X) ⊣ (−)X. When such an enrichment and left adjoint exists, we say that the functor F is an enriched coproduct. We begin our search for enriched coproducts with the category Top of topological spaces enriched in itself. We discuss several models for an enriched coproduct F(Z,X) = Z X of topological spaces indexed by another topological space, and prove some of their properties.