title: Core-compact space
text: # 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.
  
  