 The real projective plane is not simply connected

To see that the real projective plane is not simply connected space, considering the lines through origin model of the real projective plane, take a loop that starts at $(1,0,0)$ and moves along the $y=0$ great circle ends at $(-1,0,0)$.

Note that both of those points are the same, so we have a loop.

Now try to shrink it to a point.

There's just no way!