Core-compact space
In general topology and related branches of mathematics, a core-compact topological space  is a topological space whose partially ordered set of open subsets is a continuous poset. Equivalently,  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 , the set of continuous functions  has a topology such that function application is a unique continuous function from  to , 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  of a point  contains a neighborhood  of  whose closure in  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

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Further reading 
 

Topology