Core-compact space

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.
Another equivalent concrete definition is that every open neighborhood of a point contains an open neighborhood of that is way-below ; is way-below 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, 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.