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 (and only if) is locally compact. (In this article locally compact means that every point has a neighborhood base of compact neighborhoods; this is definition (3) in the linked article.)
Another equivalent concrete definition is that every open neighborhood of a point contains an open neighborhood of that is way-below ; is way-below (or relatively compact in) if and only if every open cover containing contains a finite subcover of . As a result, every locally compact space is core-compact. For Hausdorff spaces (or more generally, sober spaces), core-compact space is equivalent to locally compact. In this sense the definition is a slight weakening of the definition of a locally compact space in the non-Hausdorff case.