Tychonoff theorem for products of compact spaces

E400161

result in general topology theorem in topology

The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.

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All labels observed (6)

Label	Occurrences
Tychonoff theorem	2
Hilbert cube is compact	1
Tychonoff fixed-point theorem	1
Tychonoff theorem for products of compact spaces canonical	1
Tychonoff's theorem	1
full Tychonoff theorem	1

Statements (48)

Predicate	Object
instanceOf	result in general topology ⓘ theorem in topology ⓘ
appearsIn	standard textbooks on general topology ⓘ
assumes	each factor space is compact ⓘ
canBeProvedUsing	Alexander subbase theorem ⓘ nets ⓘ ultrafilters ⓘ
concludes	the product space is compact ⓘ
context	Zermelo–Fraenkel set theory ⓘ
doesNotRequire	axiom of choice for finite products ⓘ
domain	arbitrary products of topological spaces ⓘ compact topological spaces ⓘ
equivalentFormulation	Every family of nonempty compact sets with the finite intersection property has nonempty intersection in the product. ⓘ Every filter on a product of compact spaces has a cluster point. ⓘ Every ultrafilter on a product of compact spaces converges. ⓘ
equivalentTo	axiom of choice ⓘ surface form: axiom of choice (over ZF) Tychonoff theorem for products of compact spaces self-linksurface differs ⓘ surface form: full Tychonoff theorem
failsFor	box topology on infinite products of compact spaces ⓘ
field	set-theoretic topology ⓘ topology ⓘ
generalizes	Borel–Lebesgue theorem ⓘ surface form: Heine–Borel theorem for products of closed bounded intervals in R
historicalNote	first proved by Andrey Tychonoff in the 1930s ⓘ
holdsIn	product topology, not box topology ⓘ
implies	Cantor cube {0,1}^I is compact for any index set I ⓘ Tychonoff theorem for products of compact spaces self-linksurface differs ⓘ surface form: Hilbert cube is compact every product of compact Hausdorff spaces is compact Hausdorff ⓘ finite product of compact spaces is compact ⓘ
logicalStrength	equivalent to axiom of choice ⓘ
namedAfter	Andrey Tychonoff ⓘ
quantification	arbitrary (possibly infinite) products ⓘ
relatedConcept	axiom of choice ⓘ compactness ⓘ product space ⓘ
relatedTo	Alexander subbase theorem ⓘ Boolean prime ideal theorem ⓘ Ultrafilter lemma ⓘ
requires	some form of choice for infinite products ⓘ
role	cornerstone of general topology ⓘ standard equivalent of the axiom of choice in topology ⓘ
specialCaseOf	Tychonoff theorem for products of compact spaces self-linksurface differs ⓘ surface form: Tychonoff theorem
statement	The product of any family of compact topological spaces is compact in the product topology. ⓘ
usedIn	construction of compactifications ⓘ construction of product measures ⓘ functional analysis ⓘ measure theory ⓘ probability theory ⓘ topological algebra ⓘ
uses	product topology ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

axiom of choice → equivalentTo → Tychonoff theorem for products of compact spaces ⓘ

Schauder fixed-point theorem → relatedTo → Tychonoff theorem for products of compact spaces ⓘ

this entity surface form: Tychonoff fixed-point theorem

de Bruijn–Erdős theorem → relatedTo → Tychonoff theorem for products of compact spaces ⓘ

this entity surface form: Tychonoff's theorem

Tychonoff theorem for products of compact spaces → specialCaseOf → Tychonoff theorem for products of compact spaces self-linksurface differs ⓘ

this entity surface form: Tychonoff theorem

Tychonoff theorem for products of compact spaces → equivalentTo → Tychonoff theorem for products of compact spaces self-linksurface differs ⓘ

this entity surface form: full Tychonoff theorem

Tychonoff theorem for products of compact spaces → implies → Tychonoff theorem for products of compact spaces self-linksurface differs ⓘ

this entity surface form: Hilbert cube is compact

Banach–Alaoglu theorem → uses → Tychonoff theorem for products of compact spaces ⓘ

this entity surface form: Tychonoff theorem