TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Brouwer fixed-point theorem for the closed disk

Statement

Every continuous map $f : D^{2} \to D^{2}$ from the closed unit disk to itself has a fixed point: there is an $x \in D^{2}$ such that $f (x) = x$ .

Facts & Assumptions

Given: A continuous map $f : D^{2} \to D^{2}$ .

[L1]

Every continuous fixed-point-free map $f : D^{2} \to D^{2}$ determines a continuous retraction $D^{2} \to S^{1}$ (A fixed-point-free self-map of the disk produces a continuous retraction onto the unit circle).

[L2]

There is no continuous retraction from the closed unit disk $D^{2}$ onto its boundary circle $S^{1}$ (There is no retraction of the closed disk onto the unit circle).

Proof

technique · contradiction

1.1

Suppose that $f$ has no fixed point, so $f (x) \neq x$ for every $x \in D^{2}$ .

givenassume-contra

2.1

By [L1], the map $f$ then determines a continuous retraction $D^{2} \to S^{1}$ , contradicting [L2]. Therefore $f$ has a fixed point.

step 1.1L1L2discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Allen Hatcher, Algebraic Topology, Theorem 1.9 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 1, §6 (standard reference, not scraped)