DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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.

Uniform space in the uniform-cover formulation

Definition

For a cover $V$ of $X$ and $A \subseteq X$ , write $St (A, V)$ for the union of the members of $V$ meeting $A$ . A cover $V$ star-refines $W$ if for every $V \in V$ , $St (V, V)$ is contained in some member of $W$ .

A uniform-cover structure is a nonempty family $C$ of covers of $X$ such that a cover refined by a member of $C$ belongs to $C$ , any two members have a common refinement in $C$ , and every member has a star-refinement in $C$ . Its members are uniform covers. When $X \neq \emptyset$ , the topology it induces and its equivalence with entourages are proved in On a nonempty set, entourage uniformities and uniform-cover structures determine one another.

Uniform space in the uniform-cover formulation

Definition

Depends on

Dependency tree · 2 levels

Sources