False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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FALSE: every subset of $\mathbb{R}$ of measure zero is nowhere dense

Statement

False claim: every subset of $\mathbb{R}$ of measure zero (Measure zero (a countable cover by intervals of total length below every $\varepsilon$ ) and content zero (a finite such cover)) is nowhere dense (Nowhere dense, meager (first category), residual, and second category subsets of $\mathbb{R}$ ).

The claim confuses two different smallness conditions. Measure zero constrains the total length of a cover; nowhere density constrains the closure. A set may be covered by intervals of total length below any $\varepsilon$ and still have every real as an adherent point, and $\mathbb{Q}$ does exactly that.

Facts & Assumptions

Given: The set $\mathbb{Q}_{\mathbb{R}} \subseteq \mathbb{R}$ of rationals, that is the image of $\mathbb{Q}$ under the canonical embedding (The rationals embed densely in the reals).

[A1]

The false claim: every subset of $\mathbb{R}$ of measure zero is nowhere dense.

[L1]

$\mathbb{Q} \approx \mathbb{N}$ , so $\mathbb{Q}_{\mathbb{R}}$ is at most countable ( $\mathbb{Q}$ is countably infinite, Finite, countably infinite, countable, uncountable, The rationals embed densely in the reals).

[L2]

Every at most countable subset of $\mathbb{R}$ has measure zero (Every at most countable subset of $\mathbb{R}$ has measure zero).

[L3]

$\mathbb{Q}_{\mathbb{R}}$ is dense in $\mathbb{R}$ , that is $\overline{\mathbb{Q}_{\mathbb{R}}} = \mathbb{R}$ (Both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$ , and every nonempty open subset of $\mathbb{R}$ is uncountable, Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ ).

[L4]

A set is nowhere dense when the interior of its closure is empty; the interior of an open set is itself, and $\mathbb{R}$ is open (Nowhere dense, meager (first category), residual, and second category subsets of $\mathbb{R}$ , Interior, closure, boundary and exterior of a subset of $\mathbb{R}$ , Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen).

Refutation

technique · direct

1.1

$\mathbb{Q}_{\mathbb{R}}$ has measure zero, being at most countable by [L1] and hence null by [L2].

L1L2

1.2

$\mathbb{Q}_{\mathbb{R}}$ is not nowhere dense: its closure is $\mathbb{R}$ by [L3], and the interior of $\mathbb{R}$ is $\mathbb{R}$ itself by [L4], since $\mathbb{R}$ is an open subset of $\mathbb{R}$ ; so the interior of the closure is $\mathbb{R} \ne \varnothing$ .

L3L4

2.1

So $\mathbb{Q}_{\mathbb{R}}$ is a subset of $\mathbb{R}$ of measure zero that is not nowhere dense, and the claim [A1] fails at it; the claim is therefore false.

step 1.1step 1.2A1∎

Remarks

$\mathbb{Q}_{\mathbb{R}}$ is nonetheless meager, being a union of countably many singletons, each of which is nowhere dense ( $\mathbb{Q}$ is $F_\sigma$ , meager and not $G_\delta$ , while the irrationals are $G_\delta$ , residual and not $F_\sigma$ ). So the failure above is not a failure of topological smallness in every sense: it is exactly the failure of the one-step condition. Meagreness is the countable-union closure of nowhere density, and it is the notion under which $\mathbb{Q}$ is small.
The converse implication is also false, and needs an uncountable witness: the Smith-Volterra-Cantor set is nowhere dense and not null (FALSE: every nowhere dense subset of $\mathbb{R}$ has measure zero).
The named witness is $\mathbb{Q}$ is dense in $\mathbb{R}$ and has measure zero ↗.

Depends on

Used by

ℚ is dense in ℝ and has measure zero Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 130 results over 32 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Null set (Wikipedia) (standard reference, not scraped)
Nowhere dense set (Wikipedia) (standard reference, not scraped)
E. Zakon, Mathematical Analysis, §6.8: Baire Categories (standard reference, not scraped)
MIT 18.125, Homework 2: Measure-zero sets (standard reference, not scraped)

FALSE: every subset of R\mathbb{R}R of measure zero is nowhere dense