Baire space ω^ω
E681626
Polish space
completely metrizable space
perfect space
separable space
standard Borel space
standard example in descriptive set theory
topological space
zero-dimensional topological space
Baire space ω^ω is a fundamental topological space consisting of all infinite sequences of natural numbers with the product topology, serving as a central object in descriptive set theory and topology.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Baire space ω^ω canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7685013 — 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 ω^ω Context triple: [Alexandrov–Hausdorff theorem, involves, Baire space ω^ω]
-
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.
Tychonoff theorem for products of compact spaces
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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C.
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.
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D.
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.
-
E.
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.
- 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 ω^ω Target entity description: Baire space ω^ω is a fundamental topological space consisting of all infinite sequences of natural numbers with the product topology, serving as a central object in descriptive set theory and 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.
Tychonoff theorem for products of compact spaces
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.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Polish space
ⓘ
completely metrizable space ⓘ perfect space ⓘ separable space ⓘ standard Borel space ⓘ standard example in descriptive set theory ⓘ topological space ⓘ zero-dimensional topological space ⓘ |
| appearsIn |
Kuratowski’s theory of Borel hierarchies
NERFINISHED
ⓘ
Lusin–Novikov uniformization theorems NERFINISHED ⓘ classical results on analytic and coanalytic sets ⓘ |
| builtFrom | countable product of the discrete space ω ⓘ |
| hasBasis | set of all cylinder sets determined by finite initial segments ⓘ |
| hasBasisElement | [s] = { x ∈ ω^ω : s ⊆ x } for finite sequence s ∈ ω^{<ω} ⓘ |
| hasCardinality | continuum ⓘ |
| hasMetric | d(x,y) = 0 if x = y, otherwise 2^{-n} where n is least index with x(n) ≠ y(n) ⓘ |
| hasNoIsolatedPoints | true ⓘ |
| hasProperty |
every meager set has dense complement
ⓘ
every nonempty open set is uncountable ⓘ intersection of countably many dense open sets is dense ⓘ |
| hasTopology | product topology of the discrete topology on ω ⓘ |
| hasUnderlyingSet |
set of all functions f: ω → ω
ⓘ
set of all infinite sequences of natural numbers ⓘ |
| isBaireSpace | true ⓘ |
| isCentralObjectIn |
descriptive set theory
ⓘ
effective descriptive set theory ⓘ general topology ⓘ |
| isCompletelyMetrizable | true ⓘ |
| isHomeomorphicTo |
set of irrationals in ℝ with the subspace topology
ⓘ
space of all functions from ω to ω with the topology of pointwise convergence from discrete ω ⓘ space of all strictly increasing sequences of natural numbers ⓘ ω^ω with the Baire metric ⓘ |
| isNamedAfter | René-Louis Baire NERFINISHED ⓘ |
| isNonCompact | true ⓘ |
| isNonLocallyCompact | true ⓘ |
| isNotHomeomorphicTo | Cantor space 2^ω NERFINISHED ⓘ |
| isPerfect | true ⓘ |
| isPrototypeOf | non-σ-compact Polish spaces ⓘ |
| isSecondCountable | true ⓘ |
| isStandardBorelSpace | true ⓘ |
| isTotallyDisconnected | true ⓘ |
| isUncountable | true ⓘ |
| isUniversalFor | Polish spaces under Borel isomorphism ⓘ |
| isUniversalFor | Polish spaces under continuous open surjections ⓘ |
| isUsedToCode |
Borel sets
NERFINISHED
ⓘ
analytic sets ⓘ real numbers in descriptive set theory ⓘ |
| symbolUses | ω to denote the set of natural numbers ⓘ |
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 ω^ω Description of subject: Baire space ω^ω is a fundamental topological space consisting of all infinite sequences of natural numbers with the product topology, serving as a central object in descriptive set theory and topology.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.