Ok, if one looks at invariants like the fundamental group, the Euler characteristic or orientability, then it is immediate to see that $\mathbb R\mathbb P^2$ is not homeomorphic to $S^2$.
Is there any simple (or maybe not simple but still intersting) proof of this fact that makse no use of sophisticated invariants? (like homology, homotopy etc...)
The purpose is to teach this fact to a class without any of such tools.
It is easy to check that removing a point from the sphere gives you something that continuously retracts to a point, whereas removing a point from the real projective plane gives you something that retracts to a circle. Therefore they can't be homeomorphic.
If you don't want to work with retractions, note that removing a point from a sphere leaves you with a disk, whereas removing a point from the real projective plane leaves you with an open Mobius band.