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How can I show that $\Bbb{P}^1$ is homeomorphic to the one point compactification of $\Bbb{R}$? I'm allowed to assume that $\Bbb{P}^1$ is Hausdorff.

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up vote 5 down vote accepted

Define $f:\mathbb{R}_\infty\to \mathbb{P}^1$ by $t\mapsto [t:1]$ and $\infty \mapsto [1:0]$. You can easily verify that $f$ is a bijective continuous map. Since $\mathbb{R}_\infty$ is compact and $\mathbb{P}^1$ Hausdorff we must have that $f$ is a closed map, and thus a homeomorphism.

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