DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nowhere dense, meagre, residual, and comeagre subsets of a topological space

Definition

Let $X$ be a topological space and let $A \subseteq X$ . The set $A$ is nowhere dense when $int (\overline{A}) = \emptyset$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space). It is meagre when there is a sequence $(N_{n})_{n \in N}$ of nowhere dense subsets of $X$ with $A \subseteq ⋃_{n} N_{n}$ . It is residual, or comeagre, when $X ∖ A$ is meagre. The empty union shows that $\emptyset$ is meagre, including when $X = \emptyset$ .

Depends on

Used by

Under Dependent Choice, nowhere differentiable functions form a residual subset of C([0,1],ℝ) Corollary
FALSE: the rational numbers form a Baire space False statement
The meagre subsets of a topological space form a sigma-ideal Proposition
Equivalent forms of the Baire property Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 10 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

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)
Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)