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.

A gauge of pseudometrics and, on a nonempty set, the uniformity it generates

Definition

A gauge of pseudometrics on $X$ is a family $P$ of pseudometrics (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric). For finite $F \subseteq P$ and $ε > 0$ , put $E (F, ε) = {(x, y) : p (x, y) < ε for every p \in F}$ . If $X \neq \emptyset$ , these sets form a filter base and generate a uniformity (Filter base and the filter it generates, The upward closure of a filter base is the smallest filter containing it), called the uniformity generated by $P$ .

For an already given uniformity $U$ on $X$ , a pseudometric $p$ is uniformly continuous for $U$ when ${(x, y) : p (x, y) < ε} \in U$ for every $ε > 0$ . With this terminology, each member of a gauge is uniformly continuous for the uniformity generated by that gauge.

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Wodzicki, Uniform Structure (standard reference, not scraped)
Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)
M. Kunzinger, General Topology (standard reference, not scraped)
M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)