$\mathbb R \mathbb P^1$ 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?
2026-03-30 10:37:47.1774867067
$\mathbb R \mathbb P^1$ is homeomorphic to $S^1$. Question about the proof.
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