Euler Charcteristic of a Crosscap
The plane diagram of the real projective plane can be deformed into a sphere with the edges in the sequence abab. Removing the lower part of the surface, which is homeomorphic to a disc, we obtain the surface on the bottom left.
By closing the hole in that the edges are joined in the directions of the arrows we obtain a closed surface intersecting itself. The surface is one sided with A and B single points and all other points of the segment AB double points. The surface obtained, on the top right, is called a cross cap.
This also the Euler characteristic of a Mobius band. In fact a Mobius band can be deformed into a cross cap.