This paper develops a unified
theory of function spaces M𝒜(Y,Z) with set-open topologies, the sets in question
being the continuous images of selected classes of topological spaces 𝒜. We prove that
at least five of these function spaces are distinct and have corresponding exponential
homeomorphisms 𝜃 : M𝒜(X,M𝒜(Y,Z))≅M𝒜(X ×𝒜Y,Z) for suitably retopologized
product spaces X ×𝒜Y . Singleton spaces are normally identities with respect to
these products and so we have determined four distinct monoidal closed structures
for the category of all spaces. Conditions for the category of spaces generated by 𝒜,
i.e., the coreflective hull of 𝒜, to be cartesian closed and/or convenient are given. One
result asserts that the category of sequential spaces is the smallest convenient
category.
