Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$, and $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$ the linear form defined recursively, satisfying $\chi_n(\mathbb{1}_{P^\circ}) = (-1)^{\operatorname{dim} P}$ for every polytope $P$.
Deduce Euler's formula $\sum_F (-1)^{\operatorname{dim} F} = 1$ where $F$ ranges over the faces of $P$.