I am working on this question:
Considering the elipse: $E=x^2/4+y^2=1$ and the equivalence relation $(x_1,y_1)R(x_2,y_2)$ if and only if $x_1^2+y_1^2=x_2^2+y_2^2$.
Is $(E/R,\epsilon_1/R)$ homeomorphic to $([1,2],\epsilon_1)$? $\epsilon_1$ is the usual topology over $ \mathbb R $
The equivalence relation tells me each equivalence class is formed of 4 points with the coordinates $(\pm x,\pm y)$ over the ellipse, so it's like the ellipse gets transformed to the arc of ellipse in the first quadrant. I have already shown that the space is connected and compact.
1) First of all, is there an error is this question? Since the ellipse leaves in $\mathbb R^2$, shouldn't it be $(E/R,\epsilon_2/R)$? or is it ok because it is a line an when restricting to it, it is in some sense a one-dimensional thing (you can only go forward or backward along it) ?
2) Intuitively it looks like it is fact an homeomorphism, but I am not sure how to prove it. Any idea?
1) This depends on what the author meant by $\epsilon_1$. Either way the context is clear: the topology is supposed to be inherited from $\mathbb{R}^2$. So whether the index is $2$ or $1$, is it typo or not, is really irrelevant.
2) Assuming you've correctly concluded that only tuples of the form $(\pm x,\pm y)$ are related (which seems correct) we have
$$q:[0,2]\to E/R$$ $$q(r)= \bigg[r, \sqrt{(1-r^2/4)}\bigg]_R$$
which is a homeomorphism. Continuity is simple because the function factorizes through $E$. You can check that under the relationship this is a bijection and thus a homeomorphism (because $[0,2]$ is compact).