The Cantor Set, Baire Category, and Measure Zero in $R$ : Examples and Counterexamples

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✓ 12 results · all verified · 9 also independently AI-judged

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1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Example

ℚ is covered by open intervals of total length ε, for every ε > 0

Example

Which points of [0,1] lie in the Cantor set, read off their ternary expansions, with 1/4 worked out

Example

The intervals removed from the Smith-Volterra-Cantor set have total length 1/2, so the set cannot be covered by intervals of total length less than 1/2

Example

Baire category gives a third proof that ℝ is uncountable

Example

The Cantor function takes the value 1/2 on all of [1/3, 2/3], and its values at 1/9, 1/4 and 7/9

Example

The Cantor set contains no interval of positive length yet has no isolated point, so every connected subset of it is a single point

Counterexample

ℝ is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions

Counterexample

1/4 lies in the Cantor set and is the endpoint of no removed interval, so the endpoints do not exhaust it

Counterexample

The irrationals form a residual G_δ set that is not F_σ

Sources

The Cantor Set, Baire Category, and Measure Zero in $R$ : Examples and Counterexamples

3 · Items

ℚ is covered by open intervals of total length ε, for every ε > 0

Which points of [0,1] lie in the Cantor set, read off their ternary expansions, with 1/4 worked out

The intervals removed from the Smith-Volterra-Cantor set have total length 1/2, so the set cannot be covered by intervals of total length less than 1/2

Baire category gives a third proof that ℝ is uncountable

The Cantor function takes the value 1/2 on all of [1/3, 2/3], and its values at 1/9, 1/4 and 7/9

The Cantor set contains no interval of positive length yet has no isolated point, so every connected subset of it is a single point

The Smith-Volterra-Cantor set is nowhere dense and does not have measure zero

ℚ is dense in ℝ and has measure zero

ℚ ∩ [0,1] has measure zero and not content zero, although it is bounded

ℝ is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions

1/4 lies in the Cantor set and is the endpoint of no removed interval, so the endpoints do not exhaust it

The irrationals form a residual G_δ set that is not F_σ

The Cantor Set, Baire Category, and Measure Zero in RR: Examples and Counterexamples

3 · Items

ℚ is covered by open intervals of total length ε, for every ε > 0

Which points of [0,1] lie in the Cantor set, read off their ternary expansions, with 1/4 worked out

The intervals removed from the Smith-Volterra-Cantor set have total length 1/2, so the set cannot be covered by intervals of total length less than 1/2

Baire category gives a third proof that ℝ is uncountable

The Cantor function takes the value 1/2 on all of [1/3, 2/3], and its values at 1/9, 1/4 and 7/9

The Cantor set contains no interval of positive length yet has no isolated point, so every connected subset of it is a single point

The Smith-Volterra-Cantor set is nowhere dense and does not have measure zero

ℚ is dense in ℝ and has measure zero

ℚ ∩ [0,1] has measure zero and not content zero, although it is bounded

ℝ is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions

1/4 lies in the Cantor set and is the endpoint of no removed interval, so the endpoints do not exhaust it

The irrationals form a residual G_δ set that is not F_σ

The Cantor Set, Baire Category, and Measure Zero in $R$ : Examples and Counterexamples