Fixed-Point Property

Definition

A topological space $(X,\mathcal{T})$ has fixed-point property if any continuous function $f:X\to X$ has at least one fixed point. $\forall (f:X\to X)\in C^0,\exists x\text{s.t.}f(x)=x$

Facts

Every closed and bounded interval has the fixed-point property.

Every closed ball on ${\mathbb{R}}^n$ has the fixed-point property. (Brouwer fixed point theorem)

Fixed-point property is a Topological Invariant.