Core-compact space

In general topology and related branches of mathematics, a core-compact topological space [math]\displaystyle{ X }[/math] is a topological space whose partially ordered set of open subsets is a continuous poset. Equivalently, [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ Y }[/math], the set of continuous functions [math]\displaystyle{ \mathcal{C}(X,Y) }[/math] has a topology such that function application is a unique continuous function from [math]\displaystyle{ X \times \mathcal{C}(X, Y) }[/math] to [math]\displaystyle{ Y }[/math], 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 [math]\displaystyle{ U }[/math] of a point [math]\displaystyle{ x }[/math] contains a neighborhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ x }[/math] whose closure in [math]\displaystyle{ U }[/math] 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.

See also

• Locally compact space

Further reading

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