Theorem

Brouwer Fixed-Point Theorem

BROW-er

Topology

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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

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

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Connections

Associated With

Banach Fixed-Point Theorem

Fixed Point (Mathematics)

The theorem guarantees a fixed point exists for any continuous self-map of a nonempty compact convex set, so its conclusion is the fixed point concept applied.

Additional Source Fixed Point (Mathematics) (Wikipedia)Lead section

Kakutani Fixed-Point Theorem

Has Statement Form

Existence Theorem

Inequality

In Branch

Algebraic Topology

Additional Source Brouwer Fixed-Point Theorem (Wikipedia)First proofs subsection

Topology

Source Brouwer Fixed-Point Theorem (Wikipedia)

Proved By

L. E. J. Brouwer

Source Brouwer Fixed-Point Theorem (Wikipedia)

Sources

1. Brouwer Fixed-Point Theorem (Wikipedia)

Wikimedia Foundation

opening sentence
It states that for any continuous function f mapping a nonempty compact convex set to itself, there is a point x0 such that f(x0) = x0.
History section
the general case for continuous mappings by Brouwer in 1911
In Branch: Algebraic Topology, First proofs subsection
one of the early achievements of algebraic topology
In Group: Fixed-Point Theorems, Lead paragraph
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L.

View the Source

Fixed Point (Mathematics) (Wikipedia)

Wikimedia FoundationAssociated With: Fixed Point (Mathematics), Lead section

Quote, Associated With: Fixed Point (Mathematics), Lead section

a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation

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Brouwer Fixed-Point Theorem

Associated With

Has Statement Form

In Branch

Proved By