Theorems

Brouwer Fixed-Point Theorem

BROW-er

Topology

Citation Formats

General Reference

Mathematics Atlas. "Brouwer Fixed-Point Theorem." Accessed August 30, 2026. https://mathematics.interactiveatlas.org/theorems/brouwer-fixed-point-theorem.

APA Style

Mathematics Atlas. (n.d.). Brouwer Fixed-Point Theorem. Retrieved August 30, 2026, from https://mathematics.interactiveatlas.org/theorems/brouwer-fixed-point-theorem

BibTeX

@misc{mathematicsatlas-brouwer-fixed-point-theorem,
  author = {Mathematics Atlas},
  title = {Brouwer Fixed-Point Theorem},
  year = {2026},
  url = {https://mathematics.interactiveatlas.org/theorems/brouwer-fixed-point-theorem},
  note = {Accessed August 30, 2026}
}

The Brouwer fixed-point theorem states that for any continuous function f mapping a nonempty compact convex set to itself, there is a point x0 such that f(x0) equals x0. Among the many fixed-point theorems, Brouwer's is particularly well known for its use across numerous fields of mathematics.

Facts

Statement

Any continuous function that maps a nonempty compact convex set into itself must leave at least one point of that set fixed, unmoved by the mapping. 1

Proof Year

1911 1

Cross-Tradition Connections