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. Expanding the definition of an exponential object, this means that for any $$Y$$, the set of continuous functions $$\mathcal{C}(X,Y)$$ has a topology such that function application is a unique continuous function from $$X \times \mathcal{C}(X, Y)$$ to $$Y$$, which is given by the Compact-open topology and is the most general way to define it.

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 (weakly) locally compact space is core-compact, and every Hausdorff (or more generally, sober ) core-compact space is locally compact, so the definition is a slight weakening of the definition of a locally compact space in the non-Hausdorff case.