Let $X=\{(x,y)\in \mathbb{R}^2|x^2-y^2-1=0\}$ with the induced euclidean topology. Let's consider the equivalence relation:
$(x,y)\mathscr{R}(x',y') \iff x=\pm x', y'=y$
Let $Y=X /\mathscr{R}$ be the quotient set with the quotient topology $\tau$,
Prove that $(X,\tau)$ connected.
The solution for the connectedness part is given and reads: " Let $\pi$ be the projection on the quotient, $Y=\pi(X \cap \{x>0\})$, then Y is connected". Can someone elaborate o it? I don't get it. I guess they are trying to say that in the quotient it is enought to take one branch of the hyperbole, which is connected and then the projection of a connected set is a connected set, but shouldn't the projection consider the whole hyperbola to make the projection?, since the whole hyperbola is not connected the argument doesn't hold. I feel is not right to take just one branch just because in that way I have a connected space. I know that projecting one branch gives the same projection as projecting the whole hyperbola, but to pass connectedness to the quotient I think you need to take the whole set not part of it.
And what about compactness and Hausdorffnes?, since the hyperbola is not compact I can't say the quotient is compact
Can someone shed some light?