  We introduce the Euler-Poincar\'e's characteristic with an elementary way and
historically. We explain also why one should call Descartes-Poincar\'e
characteristic instead of the Euler-Poincar\'e's characteristic. All the
considered spaces are compact and without boundary. We work essentially on
smooth and oriented surfaces . However, we work also on singular and
non-oriented surfaces. We provide also the results which may be extended for
the case of superior dimension.

Algebraic Topology