TITLE: The space obtained by identifying the antipodal points of a circle
QUESTION [11 upvotes]: Take the unit circle $S^1$ in $\Bbb{R}^2$ and partition it into subsets which contain exactly two points, the points being antipodal (at opposite ends of a diameter). $P$ is the resulting identification space. 
Now, what is homeomorphic to $P$? My intuition says that it must be a semicircle. But couldn't construct a homeomorphism.

REPLY [6 votes]: I think your space is homeomorphic to the real projective line $\mathbb{R}P^1$.
$\mathbb{R}P^1$ is space of one dimensional subspaces of $\mathbb{R}^2$. Look at wiki page, as you can see real projective line is homeomorphic to $S^1$. So what you get after your construction is circle again.
In fact if you do your construction in more dimensions. Take $S^n$ and glue antipodal points and you get $\mathbb{R}P^n$. But for example in two dimensions $\mathbb{R}P^2$ is not homeomorphic to sphere since real projective plane has genus 1.