My professor is teaching quotient space recently and he gives very informal arguments (drawing diagrams, arrows and stuff). I think he wants students to get familiar with quotient spaces and visualize them.
This is the definition of Projective plane:
Set $x\sim x$ and $x\sim -x$ for all $x\in S^2$
Then, $\sim$ is an equivalence relation on $S^2$.
Then $P^2$ is defined as $S^2/\sim$.
Then, he gave us an assignment that is:
Let $D_1,D_2$ be open disks in $P^2$.
Paste the boundaries of $P^2\setminus D_1$ and $P^2\setminus D_2$ and call it $V$ (That is, set them a equivalence relation)
Then, $V$ is homeomorphic to the Klein Bottle.
How do I show this?
FYI, my professor define the Klein Bottle as the quotient space of $X$ where the equivalence relation is given by $(0,t)\sim (1,t)$ and $(t,0)\sim(1-t,1)$.