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.^[1][2][3] 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 neighborhood ${\displaystyle U}$ of a point ${\displaystyle x}$ contains a neighborhood ${\displaystyle V}$ of ${\displaystyle x}$ whose closure in ${\displaystyle U}$ is compact.^[1] As a result, every locally compact space is core-compact. For Hausdorff spaces (or more generally, sober spaces^[5]), 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.

See also

• Locally compact space