Core-compact space

In general topology and related branches of mathematics, a core-compact topological space ${\displaystyle X}$ is a topological space whose partially ordered set of open subsets is a continuous poset.¹ Equivalently, ${\displaystyle X}$ is core-compact if it is exponentiable in the category Top of topological spaces.¹²³ This means that the functor ${\displaystyle X\times -:{\bf {{Top}\to {\bf {Top}}}}}$ has a right adjoint. Equivalently, for each topological space ${\displaystyle Y}$, there exists a topology on the set of continuous functions ${\displaystyle {\mathcal {C}}(X,Y)}$ such that function application ${\displaystyle X\times {\mathcal {C}}(X,Y)\to Y}$ is continuous, and each continuous map ${\displaystyle X\times Z\to Y}$ may be curried to a continuous map ${\displaystyle Z\to {\mathcal {C}}(X,Y)}$. Note that this is the Compact-open topology if (and only if)⁴ ${\displaystyle X}$ 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 ${\displaystyle U}$ of a point ${\displaystyle x}$ contains an open neighborhood ${\displaystyle V}$ of ${\displaystyle x}$ that is way-below ${\displaystyle U}$; ${\displaystyle V}$ is way-below (or relatively compact in) ${\displaystyle U}$ if and only if every open cover containing ${\displaystyle U}$ contains a finite subcover of ${\displaystyle V}$.¹ As a result, every locally compact space is core-compact. For Hausdorff spaces (or more generally, sober spaces⁵), 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.