Brouwer fixed-point theorem

Every continuous map from a nonempty compact convex subset of Euclidean space to itself has a fixed point: some point maps to itself.

Sperner's lemma: A combinatorial result guaranteeing a fully labeled small simplex under specified boundary labeling rules. Its labeling argument yields a fixed point through increasingly fine triangulations.
Continuous function: A function whose values change continuously with its input, preserving limits of convergent sequences. Continuity prevents the map from evading a fixed point through a discontinuous jump.
Luitzen Egbertus Jan Brouwer: A Dutch mathematician who founded influential approaches to topology and intuitionistic mathematics. His 1911 work established the fixed-point result now bearing his name.
Nash equilibrium: A strategy profile in a game where no player can improve their payoff by changing strategy alone. Fixed-point theorems underpin existence proofs for equilibria in finite games.
Banach fixed-point theorem: A theorem stating that a contraction on a complete metric space has a unique fixed point. Unlike Brouwer's existence-only result, it requires contraction and provides uniqueness and an iterative method.
No-retraction theorem: The theorem that a closed ball cannot be continuously retracted onto its boundary sphere. A fixed-point-free map would construct precisely the forbidden retraction.
Compact space: A topological space in which every open cover has a finite subcover. Compactness supplies the finite-dimensional boundedness conditions behind the theorem's domain.
Henri Poincaré: A French mathematician whose work helped establish topology, dynamical systems, and mathematical physics. His questions about periodic orbits helped motivate fixed-point methods in topology.
Walrasian equilibrium: A market state in which supply equals demand across goods at prices that clear markets. Economic equilibrium existence proofs use fixed-point results for price-adjustment maps.
Schauder fixed-point theorem: A fixed-point theorem for continuous maps from a convex subset of a normed space into a relatively compact subset. It extends Brouwer's finite-dimensional result to certain infinite-dimensional spaces.

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