Core-compact space

In general topology and related branches of mathematics, a core-compact topological space $X$ is a topological space whose partially ordered set of open subsets is a continuous poset. Equivalently, $X$ is core-compact if it is exponentiable in the category Top of topological spaces. This means that the functor $X\times -:\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}\mathbf{\to }\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ has a right adjoint. Equivalently, for each topological space $Y$, there exists a topology on the set of continuous functions ${𝒞}(X,Y)$ such that function application $X\times {𝒞}(X,Y)\to Y$ is continuous, and each continuous map $X\times Z\to Y$ may be curried to a continuous map $Z\to {𝒞}(X,Y)$. Note that this is the Compact-open topology if (and only if) $X$ 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 $U$ of a point $x$ contains a neighborhood $V$ of $x$ whose closure in $U$ is compact. 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.

See also

• Locally compact space

Further reading

• "core-compact but not locally compact". Stack Exchange. June 20, 2016. https://math.stackexchange.com/q/1287458.