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.
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
, 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
[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.