The following is a statement in J. Vick's Homology Theory which makes me a little confused:
Let $X=D^2$ (the closed unit disk), $A=S^1$ and $Y$ be a copy of $S^1$ disjoint from $X$. Let $f\colon A\to Y$ be the map given by $f(z)=z^2$. Then the identification space $X\cup_fY$ is the real projective plane $\mathbb RP^2$.
It is easy to figure out that such an $X\cup_fY$ is homeomorphic to $D^2/\sim_1$, where $\sim_1$ is the equivalence relation of $D^2$ that identifies the antipodes on its boundary $S^1$. While $\mathbb RP^2$ is defined as the quotient space $S^2/\sim_2$ where $\sim_2$ identifies each pair of antipodes on $S^2$, it is not so obvious that these two spaces $D^2/\sim_1$ and $S^2/\sim_2$ are homeomorphic.
What I am aware of is that if a continuous surjection $D^2\to S^2/\sim_2$ that identifies the antipodes of $D^2$ (or similarly a surjection $S^2\to D^2/\sim_1$) can be figured out, then the homeomorphism is found. But I was also stuck in finding such a surjection. So I would like to ask how to prove their being homeomorphic (and I guess that more generally $\mathbb RP^n$ should be homeomorphic to $D^n/\sim$ where $\sim$ identifies the antipodes of $S^{n-1}$?). Thanks in advance...
The surjection you are looking for is the one composed of the embedding of $D^2$ as (e.g.) northern hemisphere of $S^2$ and the canonical projection $S^2\to S^2/\sim_2$.