TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

Every uniformly continuous map is continuous for the induced topologies

Statement

Every uniformly continuous map between uniform spaces is continuous for their induced topologies.

Facts & Assumptions

Given: A uniformly continuous map $f:X\to Y$ and a point $x\in X$ .

[A1]

Uniform continuity sends one source entourage into each prescribed target entourage (Uniformly continuous map between uniform spaces).

[L1]

Entourage balls are neighbourhood bases for the induced topologies (The sets containing an entourage ball about each of their points form a topology).

[L2]

A map is continuous at $x$ when every neighbourhood of $f(x)$ has a neighbourhood of $x$ mapped into it (Continuity of a map of topological spaces at a point and globally).

Proof

technique · direct

1.1

Let $N$ be a neighbourhood of $f(x)$ and choose a target entourage $V$ with $V[f(x)]\subseteq N$ .

L1choose

2.1

Uniform continuity supplies a source entourage $U$ whose pairs map into $V$ , so $f[U[x]]\subseteq V[f(x)]\subseteq N$ .

A1step 1.1

3.1

Since $U[x]$ is a neighbourhood of $x$ , [L2] gives continuity at $x$ ; as $x$ was arbitrary, $f$ is continuous.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 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

J. Wodzicki, Uniform Structure (standard reference, not scraped)