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] This means that the functor has a right adjoint. Equivalently, for each topological space , there exists a topology on the set of continuous functions such that function application is continuous, and each continuous map may be curried to a continuous map . Note that this is the Compact-open topology if (and only if) [4] is locally compact. (In this article locally compact means that every point has a neighborhood base of compact neighborhoods; this is definition (3) in the linked article.)
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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 locally compact space is core-compact. For Hausdorff spaces (or more generally, sober spaces [5] ), core-compact space is equivalent to locally compact. In this sense the definition is a slight weakening of the definition of a locally compact space in the non-Hausdorff case.
