# what is the one point compactification of R?

It is said that the one point compactification of R is a circle. But how do i show it? I know it suffices to show R is homeomorphic to a punctured circle but how can i prove it?

• You can search: Alexandroff compactification: en.wikipedia.org/wiki/Alexandroff_extension Jun 16, 2016 at 15:04
• SImply consider $\Bbb{R} \to S^1$ defined by $t \mapsto e^{2i \arctan t}$. Jun 16, 2016 at 15:14
• @Novati: Thanks i will check it up Jun 16, 2016 at 15:44
• @Crostul: That is not even a surjection Jun 16, 2016 at 15:55
• It's a surjection onto $S^1 \setminus \{ -1 \}$. Jun 16, 2016 at 16:43

Pictorially, you can imagine $$\mathbb R$$ to be a tangent of the circle at point $$x \in S^1$$. Now for each point $$y \in \mathbb R$$ imagine a segment joining $$y$$ and $$-x$$; $$z$$ be the point of intersection of this segment and the circle. The correspondence $$y \mapsto z$$ is a homeomorphism between $$\mathbb R$$ and $$S^1 - \{-x\}$$. • this is a quite old answer, but nonetheless; shouldn't it read, for any point $y\in\mathbb R$ instead of $y\in S^1$? Mar 11, 2020 at 9:50