TITLE: what is the one point compactification of R?
QUESTION [5 upvotes]: 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?

REPLY [8 votes]: 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 $S^1 - \{-x\}$.
The nice thing about this construction is that it is generalizable!