Recall that $P^2$ can be obtained by attaching a 2 cell to $P^1$. Denote this 2 cell by $B_2$. Now remove an open disk from $B_2$, according to this post it should be the Mobius strip. However it seems to me that it should be $P^1$.
My reasoning: With a disk removed, we can deform $B_2$ to its boundary. Using the description of $P^2$ at the beginning, we then conclude that $P^2$ with a disk removed deformation retracted onto $P^1$.