## Brouwer fixed point theorem

Let be an open subset of such that is homeomorphic to the closed unit ball ( in ). If (continuous function on ) and , then has a fixed point in .

**Proof: **First reduce the problem to the case where . To do this, consider the homeomorphism . Then maps to , and implies (since there exists such that ) we have , which means . This reduces the problem to consider our domain the unit ball in .

Now, if for some then we are done. Suppose this does not happen. Consider the homotopy . It is clear that if and , then , hence . By hypothesis, . By invariance of topological degree of a mapping at a point which is not in the image of the boundary of the open set on which the mappings are is defined, under a homotopy, we have ( is the identity mapping). But since .

If the topology degree of a mapping at a point is not , then there exists a point in the domain of such . Therefore, there exists an element such that , meaning that , and therefore has a fixed point.