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Brouwer Fixed-Point Theorem

1911

L. E. J. Brouwer

This theorem states that for any continuous function \(f\) mapping a compact convex set to itself, there is a point \(x_0\) such that \(f(x_0) = x_0\). This point is called a fixed point. Informally, if you take a map of a country, crumple it up, and place it inside the country’s borders, there will always be at least one point on the map directly above its corresponding real-world location.

The Brouwer fixed-point theorem is a cornerstone of fixed-point theory and has profound implications in many areas of mathematics. The theorem applies to any continuous function \(f: D^n \to D^n\), where \(D^n\) is the closed n-dimensional unit ball. The proof is non-constructive; it guarantees the existence of a fixed point but does not provide a method to find it. The proof for \(n=1\) is a simple consequence of the Intermediate Value Theorem. For higher dimensions, the proof is more complex and typically relies on tools from algebraic topology, such as homology or the concept of the degree of a map. One common proof strategy uses a retraction argument. It assumes, for the sake of contradiction, that a continuous function \(f: D^n \to D^n\) has no fixed point. One can then construct a continuous function (a retraction) \(r: D^n \to S^{n-1}\) from the disk to its boundary sphere, which can be shown to be impossible. The theorem’s power lies in its generality; it requires only continuity of the function and compactness and convexity of the domain, making it applicable to a wide range of problems where one needs to prove the existence of a solution or equilibrium state.

UNESCO Nomenclature: 1209

– Topology