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

How this entity was disambiguated

This entity first appeared as the object of triple T3913048 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Tychonoff theorem for products of compact spaces
Context triple: [axiom of choice, equivalentTo, Tychonoff theorem for products of compact spaces]

A. Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
B. Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
C. Tarski’s fixed point theorem
Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
D. Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
E. Hamilton’s compactness theorem
Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Tychonoff theorem for products of compact spaces
Target entity description: 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.

A. Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
B. Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
C. Tarski’s fixed point theorem
Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
D. Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
E. Hamilton’s compactness theorem
Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
F. None of above. chosen

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 ⓘ

How these facts were elicited

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 ⓘ