Existing in Mathlib · Topology

Heine–Cantor Theorem

A continuous map from a compact uniform space into any uniform space is automatically uniformly continuous.

compact spaces
uniform spaces
continuous maps
uniform continuity
uniformity filters
diagonal neighborhoods

Exact theorem

Exact Mathlib statement

theorem CompactSpace.uniformContinuous_of_continuous {α β : Type*} [UniformSpace α] [UniformSpace β] [CompactSpace α] {f : α → β} (h : Continuous f) : UniformContinuous f

The theorem at a glance

Heine–Cantor theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory Heine–Cantor poster places a compact paired domain field and its diagonal neighborhoods on the left, sends both coordinates through f, and lands in diagonal neighborhoods of a target uniform space on the right beneath the continuity-to-uniform-continuity statement. — On a compact uniform-space domain, ordinary continuity already controls all sufficiently close pairs by one uniform rule.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

An enclosed ivory source field holds eight shape-matched point pairs beside a bright diagonal seam; paired strands pass through a central two-ring aperture into eight nested target halos around a second seam. — One compactly controlled source neighborhood carries every shown nearby pair through the same map into target diagonal neighborhoods.

Why it matters

A mathematical landmark

Heine–Cantor is a central compactness principle in analysis and topology: a pointwise local continuity condition becomes one uniform condition across the entire domain. Mathlib's compact abstract proof makes the bridge especially clear by expressing uniformity through neighborhoods of the diagonal.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's whole-space CompactSpace form of the Heine–Cantor theorem: for uniform spaces α and β, a CompactSpace instance on α, and a map f : α → β, continuity of f implies uniform continuity. It is not the nearby compact-subset theorem for ContinuousOn and UniformContinuousOn, it does not assert a converse, and it does not specialize the abstract uniform-space statement to metric spaces or intervals.

The selected declaration concerns the whole compact domain through a CompactSpace instance; it is not the nearby compact-subset theorem using ContinuousOn and UniformContinuousOn.
The theorem proves that continuity implies uniform continuity under compactness; it does not assert the converse.
The declaration is abstract over uniform spaces and adds no metric, Hausdorff, separatedness, interval, epsilon-delta modulus, or quantitative Lipschitz hypothesis.
ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: CompactSpace.uniformContinuous_of_continuous in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.heine-cantor-theorem.v001
Source: Open the pinned upstream reference