$\mathbb R \mathbb P^1$ is homeomorphic to $S^1$ by identifying antipodal points. This is pretty clear to me. But my question is: Why are we allowed to identify antipodal points? Because if we do so, then we no longer have $S^1$ as the homeomorphic space, but a quotient space. Am I wrong?
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3What is $\Bbb RP^1$? – Ted Shifrin Jul 14 '23 at 21:19
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1Why would you assume that $S^{1}$, a circle is hoemeomorphic to $\Bbb{RP}^{1}$? Ofcourse $\Bbb{RP}^{1}$ is homeomorphic to the quotient and not $S^{1}$. In fact many books define $\Bbb{RP}^{1}$ in that way. – Mr.Gandalf Sauron Jul 14 '23 at 21:19
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4This quotient of $S^1$ is indeed homeomorphic to $S^1.$ – Anne Bauval Jul 14 '23 at 21:23
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1Here is a good way to see it in my opinion: https://math.stackexchange.com/a/4730545/1101015 – Snared Jul 14 '23 at 21:34
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Your question "Why are we allowed to identify antipodal points?" is too imprecise and lacking to much context to be answerable. – Lee Mosher Jul 14 '23 at 22:04
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1Do you think that just because the real line is homeomorphic to itself, that it cannot be homeomorphic to the interval (0, ∞) ? – Dan Asimov Jul 15 '23 at 00:05