Bernstein set
E354908
A Bernstein set is a subset of the real numbers that intersects every uncountable closed set yet contains none of them, serving as a classic example of a non-measurable, highly pathological set in set theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernstein set canonical | 2 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
pathological subset of the real line
ⓘ
set-theoretic concept ⓘ subset of the real numbers ⓘ |
| cannotBe |
F_sigma subset of R
ⓘ
G_delta subset of R ⓘ co-countable ⓘ countable ⓘ |
| cardinalityProperty | both the set and its complement have cardinality continuum ⓘ |
| complementProperty | complement is also a Bernstein set ⓘ |
| constructionMethod | transfinite recursion using the axiom of choice ⓘ |
| definedOn |
Cantor set
ⓘ
surface form:
Cantor space
real numbers ⓘ |
| field |
real analysis
ⓘ
set theory ⓘ |
| hasProperty |
cardinality continuum
ⓘ
contains no perfect subset ⓘ contains no uncountable closed subset of the real line ⓘ dense in every uncountable closed subset of the real line in the sense of nonempty intersection ⓘ has no Baire property ⓘ intersects every uncountable closed subset of the real line ⓘ non-measurable with respect to Lebesgue measure ⓘ not Borel ⓘ not Lebesgue measurable ⓘ not analytic ⓘ not coanalytic ⓘ |
| intersectionProperty | meets every uncountable closed subset of R in at least one point ⓘ |
| logicalStatus | existence provable in ZFC ⓘ |
| namedAfter | Felix Bernstein ⓘ |
| relatedConcept |
Baire property
ⓘ
Borel set ⓘ Cantor set ⓘ Lebesgue measure ⓘ Vitali set ⓘ analytic set ⓘ coanalytic set ⓘ non-measurable set ⓘ perfect set ⓘ |
| requiresAxiom | axiom of choice for existence proof ⓘ |
| subsetOf |
Polish spaces via homeomorphism to the real line
ⓘ
real line ⓘ |
| topologicalProperty |
not Borel measurable
ⓘ
not F_sigma ⓘ not G_delta ⓘ |
| usedAs |
counterexample in measure theory
ⓘ
counterexample in topology ⓘ example in descriptive set theory ⓘ example of a non-measurable set ⓘ |
| yearIntroduced | 1908 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.