In general topology and related branches of mathematics, a core-compact topological space X {\displaystyle X} is a topological space whose partially ordered set of open subsets is a continuous poset. Equivalently, X {\displaystyle X} is core-compact if it is exponentiable in the category Top of topological spaces. This means that the functor
X × − : T o p → T o p {\displaystyle X\times -:{\bf {{Top}\to {\bf {Top}}}}}
has a right adjoint. Equivalently, for each topological space Y {\displaystyle Y} , there exists a topology on the set of continuous functions
C ( X , Y ) {\displaystyle {\mathcal {C}}(X,Y)} such that function application
X × C ( X , Y ) → Y {\displaystyle X\times {\mathcal {C}}(X,Y)\to Y} is continuous, and each continuous map
X × Z → Y {\displaystyle X\times Z\to Y} may be curried to a continuous map
Z → C ( X , Y ) {\displaystyle Z\to {\mathcal {C}}(X,Y)} . Note that this is the Compact-open topology if (and only if)
X {\displaystyle X} 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 U {\displaystyle U} of a point x {\displaystyle x} contains an open neighborhood V {\displaystyle V} of x {\displaystyle x} that is way-below U {\displaystyle U} ; V {\displaystyle V} is way-below (or relatively compact in) U {\displaystyle U} if and only if every open cover containing U {\displaystyle U} contains a finite subcover of V {\displaystyle V} . 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.
References
Further reading "core-compact but not locally compact". Stack Exchange. June 20, 2016.