CW complex structure of $\mathbb{CP}^1$

Question

I'm trying to prove that $\mathbb{CP}^1$ has a CW complex structure.

The problem is that I'm confused about the formal definition of a CW complex (although Hatcher's intuitive explanations make perfect sense).

I know what the structure is supposed to be: $\mathbb{CP}^1$ is diffeomorphic to $\mathbb{S}^2$, which is the attachment of a $2$-cell to a point, so I should think of attaching $U_1:=\{(z_0:z_1)\in\mathbb{CP}^1\mid z_1\neq 0\}$ (i.e., $\mathbb{S}^2$ minus the north pole) to the point at infinity $(1:0)$ (north pole).

I've constructed the map $\phi:\mathbb{D}^2\to \mathbb{CP}^1$ with $z\mapsto (z:\sqrt{1-|z|^2})$, which maps the interior of $\mathbb{D}^2$ to $U_1$ and the boundary $\partial\mathbb{D}^2$ to $(1:0)$.

This looks exactly like what I need. But is this enough? If not, what else do I need to prove?

Answer 1

Well if you know that $S^2\cong \mathbb C P^1$ you only need a CW-structure on $S^2$.

For that you need a homeomorphism $D^2/S^1\to S^2$ and a proof that the following square is a pushout (where $N\in S^2$ is, say, the north pole) :

$$\require{AMScd} \begin{CD} S^1 @>>> \{N\} \\ @VVV @VVV\\ D^2 @>>> S^2 \end{CD}$$

The proof that it is a pushout is the same as the one that $D^2/S^1$ is homeomorphic to $S^2$.

(note that since everything in sight is compact, and $S^2$ is Hausdorff this makes things easier : you just need to define a continuous bijection $D^2/S^1\to S^2$)