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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 $P^1$.

glS
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koch
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    Do you not believe a Moebius strip deformation retracts to its centerline, taking its boundary to to that centerline twice (which sounds much like $P^1$...)? – Eric Towers May 22 '19 at 02:04
  • Oh, right! So it does deformation retract onto $\mathbb RP^1$. – koch May 22 '19 at 02:22

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