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