Source: cirosantilli/the-real-projective-plane-is-not-simply-connected
= 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 (topology)> 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!