TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Brouwer fixed point theorem

Statement

Every continuous map $f : D^{n} \to D^{n}$ of the closed unit ball has a fixed point, for every integer $n \geq 0$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

For every integer $n \geq 1$ , there is no continuous retraction $D^{n} \to S^{n - 1}$ of the closed unit ball onto its boundary. (No retraction from a disk onto its boundary)

[F2]

For $n \geq 1$ , a continuous fixed-point-free map $f : D^{n} \to D^{n}$ would produce a continuous retraction $r : D^{n} \to S^{n - 1}$ by following the ray from $f (x)$ through $x$ to the boundary. (A fixed point free ball map produces a boundary retraction)

Proof

1.1given

When $n = 0$ , $D^{0}$ is a singleton, and its unique point is fixed by every self-map.

2.1F1F2∎

For $n \geq 1$ , a fixed-point-free map would give a continuous boundary retraction by F2. F1 excludes precisely such a retraction in every positive dimension. Therefore a fixed point exists.

Depends on

Dependency tree · 2 levels

Sources

Miller, Algebraic Topology I lecture notes, Theorem 10.7 and proof, p.24