In this exercise, we prove a generalization of Brouwer's Fixed Point Theorem to compact sets that are not necessarily convex, but are homeomorphic to a convex set.

Two compact sets X and Y in R_n are called homeomorphic if there exists a continuous bijection g : X → Y satisfying the property that g^-1 is also continuous.

Prove that if X ⊆ R_n is a compact set that is homeomorphic to a convex and compact set Y and if f : X → X is a continuous function, then f has a fixed point.