Disc quotient that is homeomorphic to the pinched torus

Question

I apologize for my previous post. There was a mistake. I want to write a quotient of the disc $D^2:={\{z\in\mathbb R^2;\parallel z\parallel \leq 1 }\}$ by an equivalence relation which is homeomorphic to the pinched torus. I have found something a little bit complicated. Some idea? Thank you.

Answer 1

The easiest way in my opinion is to say that the pinched torus is (homeomorphic to) $D^2/\sim$ where $$ x \sim y \Leftrightarrow \text{Either of }\cases{||x|| = ||y|| = 1\\||x|| = 1 \text{ and } y = 0\\ x = 0 \text{ and } ||y|| = 1} $$ The first condition will collapse $S^1 = \{z \in D^2 \mid ||z|| = 1\}$ to one point, making $D^2$ into the sphere $S^2$, and the two next ones will glue the south pole and north pole of that sphere together to make a pinched torus.