$\mathbb C P^1\cong (D^2\times\{1\}+ D^2\times \{-1\})\big /_\sim $

Question

I need to proove a bigger result that $\mathbb C P^1$ is homeomorphic to $S^2$.
For that I have already showed $$S^2\cong (D^2\times\{1\}+ D^2\times \{-1\})\big /_\sim$$ where $(z,1)\sim (z,-1) \iff z \in S^1$ and identity anywhere else.
Now I'm searching for two subsets in $\mathbb C P^1$ which union is homeomorphic to $D^2\times\{1\}+D^2\times \{-1\}\big /_\sim$.
I tried to partition lines $[a:b]$ in $\mathbb C^2$ in such sets where either $[a:b]=[1:c]$ or $[a:b]=[0:d]$, but second one is to "small" to be homeomorphic.
I need some hints guys.

Answer 1

Your method does work.

Take $$ D^2_{\rm first} = \{ [1 : z] \in \mathbb {CP}^1 : | z | \leq 1 \}$$ $$ D^2_{\rm second} = \{ [w : 1] \in \mathbb {CP}^1 : | w | \leq 1 \}$$ and identify the boundary points like this: $$[1 :e^{i\theta}] \in D^2_{\rm first} \ \sim \ [ e^{-i\theta} : 1] \in D^2_{\rm second} \ \ \ \ {\rm for \ all \ } \theta \in [0, 2\pi) $$

It is then immediate that $$ \mathbb {CP}^1 \cong \left( D^2_{\rm first} \coprod D^2_{\rm second} \right) / \sim \ \cong \ S^2. $$