Brouwer's fixed-point theorem

Brouwer's fixed-point theorem states that any continuous function mapping a compact convex set into itself has at least one fixed point — a point that the function maps to itself. Proved by L. E. J. Brouwer in 1910, before his foundational revolution in intuitionism, the theorem is one of the cornerstones of algebraic topology. It has profound applications in economics (proving the existence of general economic equilibrium), game theory (the existence of Nash equilibria), and differential equations. The theorem is non-constructive: it guarantees the existence of a fixed point without providing a method for finding it, a feature that makes it controversial from a constructivist perspective and connects the theorem to deeper questions about the meaning of existence proofs in mathematics.