TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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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.

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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 · two levels

8 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)