In general topology and related branches of mathematics, a core-compact topological space is a topological space whose partially ordered set of open subsets is a continuous poset. Equivalently, is core-compact if it is exponentiable in the category Top of topological spaces. This means that the functor has a right adjoint. Equivalently, for each topological space , there exists a topology on the set of continuous functions such that function application is continuous, and each continuous map may be curried to a continuous map . Note that this is the Compact-open topology if is locally compact.