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!