In M Nakahara's Geomertry, Topology and Physics, he states (example 2.4) that the the quotient space(set of equivalence classes) of the real line under the equivalence relation:
$x$ is related to $y$ if $x-y=2n\pi$, $n$ is an integer.
is the circle $S^1$. I can see why that is so. For instance, in the quotient space, $0,2\pi,4\pi$ are all in the same element(in same equivalence class); this is also true for the circle. However, this is also true for the ellipse, or for any closed curve with the perimeter $2\pi$. Why does the quotient space have to be just a circle then?
In topology, there is no distinction between a circle and an ellipse. They are both $S^1$. The distinction only appears when you have a metric and start measuring distances and curvature.