Connectedness, compactness and Hausdorffness on quotient topology with equivalence relation: $(x,y)\mathscr{R}(x',y') \iff x=\pm x', y'=y$

Question

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?

Answer 1

The positive branch of the hyperbola is a connected set, and $Y$ is the image of that set under the continuous map $\pi$, so it must be connected: continuous maps take connected sets to connected sets. What they do to disconnected sets is unpredictable. Sometimes, as is the case here, they take a disconnected set — the whole hyperbola — to a connected set. Sometimes, as in the case of a constant function, they send every disconnected set to a connected set. And sometimes, as in the case of the identity function on any space, they take all disconnected sets to disconnected sets. None of this changes the fact that the image of a connected set under a continuous map is always connected.

For the rest, you can check that the restriction of $\pi$ to the positive branch of the hyperbola is actually a homeomorphism, so $Y$ is Hausdorff but not compact.