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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.