Definition
κ is a cardinality.
A subset C of a topological space X is κ-hypercompact iff there is a collection Y of open subsets with the cardinality strictly smaller than κ such that the open sets not containing C are exactly those which are contained in some member of Y.
Remark
A subset of a space is supercompact (Definition 2) iff it is nonempty and 2-hypercompact.
Reference
Marcel Erne, Infinite distributive laws versus local connectedness and compactness properties, Topology and its Applications 156 (2009) 2054-2069.