Supercompact

Supercompact is a term used in mathematics to describe a property that expresses extreme strength of a notion of compactness or size. In different areas, it has related but distinct meanings. In set theory, a cardinal κ is supercompact if, for every λ ≥ κ, there exists an elementary embedding j: V → M with critical point κ such that j(κ) > λ and M includes all λ-sequences; equivalently, there is a normal fine κ-complete ultrafilter on P_κ(λ) for every λ ≥ κ.

Supercompact cardinals are among the strongest large cardinal hypotheses and imply the consistency of many other strong axioms; their existence cannot be proven from ZFC if ZFC is consistent. The notion is central to studies of the foundations of mathematics, with various equivalent formulations in terms of ultrafilters and elementary embeddings.

In topology, a space is called supercompact if it strengthens compactness via a covering property: every open cover has a refinement with a particularly strong structure. The precise formulations are technical and are studied in the context of compact spaces, dyadic spaces, and continuum theory.

The term thus marks a high level of “largeness” or “strength” within its respective framework, and it connects to core concepts such as ultrafilters, elementary embeddings, and the landscape of large cardinal hypotheses in set theory.