Baire space

E681622

Baire space (in the sense of Baire category) Hausdorff space Polish space T0 space T1 space Tychonoff space completely metrizable space perfect space separable space topological space zero-dimensional topological space

Baire space is a fundamental topological space—typically the set of all infinite sequences of natural numbers with the product topology—that serves as a central object in descriptive set theory and general topology.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (3)

Label	Occurrences
Baire space canonical	1
Baire spaces	1
Cantor space	1

How this entity was disambiguated

This entity first appeared as the object of triple T7684990 — 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: Baire space
Context triple: [Alexandrov–Hausdorff theorem, concerns, Baire space]

A. Baire category theorem
The Baire category theorem is a fundamental result in topology and functional analysis stating that complete metric (or locally compact Hausdorff) spaces cannot be written as countable unions of nowhere dense sets, with powerful consequences for the structure of such spaces.
B. Banach–Mazur game
The Banach–Mazur game is an infinite two-player topological game used to characterize properties such as Baire category and completeness in metric and topological spaces.
C. Lindelöf space
A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
D. Stone–Čech compactification
The Stone–Čech compactification is a construction in topology that associates to any topological space a universal, maximally extensive compact Hausdorff space into which it densely embeds.
E. Hausdorff
Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
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: Baire space
Target entity description: Baire space is a fundamental topological space—typically the set of all infinite sequences of natural numbers with the product topology—that serves as a central object in descriptive set theory and general topology.

A. Baire category theorem
The Baire category theorem is a fundamental result in topology and functional analysis stating that complete metric (or locally compact Hausdorff) spaces cannot be written as countable unions of nowhere dense sets, with powerful consequences for the structure of such spaces.
B. Banach–Mazur game
The Banach–Mazur game is an infinite two-player topological game used to characterize properties such as Baire category and completeness in metric and topological spaces.
C. Lindelöf space
A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
D. Stone–Čech compactification
The Stone–Čech compactification is a construction in topology that associates to any topological space a universal, maximally extensive compact Hausdorff space into which it densely embeds.
E. Hausdorff
Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
F. None of above. chosen

Statements (51)

Predicate	Object
instanceOf	Baire space (in the sense of Baire category) ⓘ Hausdorff space ⓘ Polish space ⓘ T0 space ⓘ T1 space ⓘ Tychonoff space ⓘ completely metrizable space ⓘ perfect space ⓘ separable space ⓘ topological space ⓘ zero-dimensional topological space ⓘ
hasBaseField	natural numbers ⓘ
hasBasisOf	cylinder sets determined by finite initial segments ⓘ
hasCardinality	continuum ⓘ
hasClopenBasis	true ⓘ
hasIsolatedPoints	false ⓘ
hasStandardMetric	d(x,y)=2^{-n} where n is first index with x(n)≠y(n) ⓘ
hasTopology	product topology ⓘ
hasTypicalElement	infinite sequence of natural numbers ⓘ
hasUnderlyingSet	N^N ⓘ set of all infinite sequences of natural numbers ⓘ
isBaireSpaceInCategorySense	true ⓘ
isCompletelyMetrizable	true ⓘ
isCompleteUnder	standard ultrametric ⓘ
isDenotedBy	N^N ⓘ ω^ω ⓘ
isHomeomorphicTo	irrational numbers with the usual topology ⓘ set of all functions from N to N with product topology ⓘ
isNamedAfter	René-Louis Baire ⓘ
isNot	compact ⓘ locally compact ⓘ σ-compact ⓘ
isNowhereLocallyCompact	true ⓘ
isPerfect	true ⓘ
isPolish	true ⓘ
isProductOf	countable discrete space of natural numbers ⓘ
isSeparable	true ⓘ
isStandardBorelSpace	true ⓘ
isStandardExampleOf	non-σ-compact Polish space ⓘ
isTotallyDisconnected	true ⓘ
isUncountable	true ⓘ
isUniversalFor	Polish spaces via continuous open surjections ⓘ
isUniversalFor	standard Borel spaces via Borel isomorphisms ⓘ
isUsedToCode	countable structures ⓘ real numbers in descriptive set theory ⓘ
isZeroDimensional	true ⓘ
playsCentralRoleIn	descriptive set theory ⓘ effective descriptive set theory ⓘ general topology ⓘ recursion theory on reals ⓘ
satisfies	Baire category theorem ⓘ

How these facts were elicited

Referenced by (3)

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

Alexandrov–Hausdorff theorem → concerns → Baire space ⓘ

Baire category theorem → concerns → Baire spaces ⓘ

linked to: Baire space

Martin-Löf randomness → appliesTo → Cantor space ⓘ