= Real projective line
{title2=$\mathbb{RP}^1$}
The real projective line $\mathbb{RP}^1$ consists of one-dimensional <vector subspaces> of $\mathbb R^2$, written $[x:y]$ with $(x,y)\ne(0,0)$ and $[x:y]=[tx:ty]$ for every nonzero real $t$. The chart $y\ne0$ gives the affine coordinate $x/y$, while $[1:0]$ is one added point at infinity.
A unit vector $(\cos\theta,\sin\theta)$ and its negative determine the same line, so the angular coordinate is taken modulo $\pi$. This quotient is itself a <circle>: the map sending that line to $(\cos2\theta,\sin2\theta)$ is well defined and gives a homeomorphism with $S^1$.
Hence the real projective line is compact and can be viewed as the one-point compactification of the real line. Both $+\infty$ and $-\infty$ approach the same added projective point, unlike the two distinct endpoints of an extended closed real interval.
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