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. [1] Equivalently, is core-compact if it is exponentiable in the category Top of topological spaces. [1] [2] [3] 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. [4]
Another equivalent concrete definition is that every neighborhood of a point contains a neighborhood of whose closure in is compact. [1] As a result, every (weakly) locally compact space is core-compact, and every Hausdorff (or more generally, sober [4]) 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.
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. [1] Equivalently, is core-compact if it is exponentiable in the category Top of topological spaces. [1] [2] [3] 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. [4]
Another equivalent concrete definition is that every neighborhood of a point contains a neighborhood of whose closure in is compact. [1] As a result, every (weakly) locally compact space is core-compact, and every Hausdorff (or more generally, sober [4]) 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.