Editing Compact space (section)

=== Algebraic examples ===

* [[Topological group]]s such as an [[orthogonal group]] are compact, while groups such as a [[general linear group]] are not.
* Since the [[p-adic numbers|{{mvar|p}}-adic integers]] are [[homeomorphic]] to the Cantor set, they form a compact set.
* Any [[global field]] ''K'' is a discrete additive subgroup of its [[adele ring]], and the quotient space is compact.  This was used in [[John Tate (mathematician)|John Tate]]'s [[Tate's thesis|thesis]] to allow [[harmonic analysis]] to be used in [[number theory]].
* The [[spectrum of a ring|spectrum]] of any [[commutative ring]] with the [[Zariski topology]] (that is, the set of all prime ideals) is compact, but never [[Hausdorff space|Hausdorff]] (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact [[scheme (mathematics)|schemes]], "quasi" referring to the non-Hausdorff nature of the topology.
* The spectrum of a [[Boolean algebra]] is compact, a fact which is part of the [[Stone representation theorem]]. [[Stone space]]s, compact [[totally disconnected space|totally disconnected]] Hausdorff spaces, form the abstract framework in which these spectra are studied.  Such spaces are also useful in the study of [[profinite group]]s.
* The [[structure space]] of a commutative unital [[Banach algebra]] is a compact Hausdorff space.
* The [[Hilbert cube]] is compact, again a consequence of Tychonoff's theorem.
* A [[profinite group]] (e.g. [[Galois group]]) is compact.