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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Baire space canonical | 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Baire space Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.