Editing Compact space (section)

== Examples ==

* Any [[finite topological space]], including the [[empty set]], is compact.  More generally, any space with a [[finite topology]] (only finitely many open sets) is compact; this includes in particular the [[trivial topology]].
* Any space carrying the [[cofinite topology]] is compact.
* Any [[locally compact]] Hausdorff space can be turned into a compact space by adding a single point to it, by means of [[Alexandroff one-point compactification]]. The one-point compactification of <math>\mathbb{R}</math> is homeomorphic to the circle {{math|'''S'''<sup>1</sup>}}; the one-point compactification of <math>\mathbb{R}^2</math> is homeomorphic to the sphere {{math|'''S'''<sup>2</sup>}}. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
* The [[right order topology]] or [[left order topology]] on any bounded [[totally ordered set]] is compact. In particular, [[Sierpiński space]] is compact.
* No [[discrete space]] with an infinite number of points is compact. The collection of all [[singleton (mathematics)|singletons]] of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
* In <math>\mathbb{R}</math> carrying the [[lower limit topology]], no uncountable set is compact.
* In the [[cocountable topology]] on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not [[locally compact]] but is still [[Lindelöf space|Lindelöf]].
* The closed [[unit interval]] {{math|{{closed-closed|0, 1}}}} is compact. This follows from the [[Heine–Borel theorem]]. The open interval {{open-open|0, 1}} is not compact: the [[open cover]] <math display="inline">\left( \frac{1}{n}, 1 - \frac{1}{n} \right)</math> for {{math|1={{mvar|n}} = 3, 4, ... }} does not have a finite subcover. Similarly, the set of ''[[rational numbers]]'' in the closed interval {{closed-closed|0,1}} is not compact: the sets of rational numbers in the intervals <math display="inline">\left[0, \frac{1}{\pi} - \frac{1}{n}\right]\text{ and }\left[\frac{1}{\pi} + \frac{1}{n}, 1\right]</math> cover all the rationals in [0,&nbsp;1] for {{math|1={{mvar|n}} = 4, 5, ... }} but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of&nbsp;<math>\mathbb{R}</math>.
* The set <math>\mathbb{R}</math> of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals {{math|{{open-open|{{mvar|n}} − 1, {{mvar|n}} + 1}} }}, where {{mvar|n}} takes all integer values in {{math|'''Z'''}}, cover <math>\mathbb{R}</math> but there is no finite subcover.
* On the other hand, the [[extended real number line]] carrying the analogous topology ''is'' compact; note that the cover described above would never reach the points at infinity and thus would ''not'' cover the extended real line.  In fact, the set has the [[homeomorphism]] to [−1,&nbsp;1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
* For every [[natural number]] {{mvar|n}}, the [[n-sphere|{{mvar|n}}-sphere]] is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional [[normed vector space]] is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its [[closed unit ball]] is compact.
* On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. ([[Alaoglu's theorem]])
* The [[Cantor set]] is compact. In fact, every compact metric space is a continuous image of the Cantor set.
* Consider the set {{mvar|K}} of all functions {{math|''f'' : <math>\mathbb{R}</math> → [0, 1]}} from the real number line to the closed unit interval, and define a topology on {{mvar|K}} so that a sequence <math>\{f_n\}</math> in {{mvar|K}} converges towards {{math|''f'' ∈ ''K''}} if and only if <math>\{f_n(x)\}</math> converges towards {{math|''f''(''x'')}} for all real numbers {{mvar|x}}. There is only one such topology; it is called the topology of [[pointwise convergence]] or the [[product topology]]. Then {{mvar|K}} is a compact topological space; this follows from the [[Tychonoff theorem]].
* A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded ([[Arzelà–Ascoli theorem]]).
* Consider the set {{mvar|K}} of all functions {{math|''f''&nbsp;: {{closed-closed|0, 1}}&nbsp;→ {{closed-closed|0, 1}}}} satisfying the [[Lipschitz condition]] {{math|{{mabs|''f''(''x'')&nbsp;−&nbsp;''f''(''y'')}}&nbsp;≤ {{mabs|''x''&nbsp;−&nbsp;''y''}}}} for all {{math|''x'',&nbsp;''y''&nbsp;∈&nbsp;{{closed-closed|0,1}}}}.  Consider on {{mvar|K}} the metric induced by the [[uniform convergence|uniform distance]] <math>d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|.</math> Then by the Arzelà–Ascoli theorem the space {{mvar|K}} is compact.
* The [[spectrum of an operator|spectrum]] of any [[bounded linear operator]] on a [[Banach space]] is a nonempty compact subset of the [[complex number]]s <math>\mathbb{C}</math>. Conversely, any compact subset of <math>\mathbb{C}</math> arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space [[sequence spaces#ℓp spaces|<math>\ell^2</math>]] may have any compact nonempty subset of <math>\mathbb{C}</math> as spectrum.
* The space of Borel [[probability measure]]s on a compact Hausdorff space is compact for the [[vague topology]], by the Alaoglu theorem.
* A collection of probability measures on the Borel sets of Euclidean space is called ''tight'' if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures.  Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.

=== 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.