I am trying to prove that if $p_1$ is the canonical projection $p_1 : \mathbb{S}^{2n+1} \longrightarrow \mathbb{CP}^n$ then the adjunction space $\mathbb{B}_{2n+2} \cup_{p_{1}} \mathbb{CP}^n$ is homeomorphic to $\mathbb{CP}^{n+1}$.
i considered $\phi : \mathbb{B}_{2n+2} \dot{\cup} \mathbb{CP}^n \longrightarrow \mathbb{CP}^{n+1}$ defined as
- $ \phi(z) = [z_1,...,z_n, \sqrt{1- \sum_{i=1}^n|z_i|^2}] $ if $z \in \mathbb{B}_{2n+2}$
- $ \phi([z]) = [z_1,...,z_n, 0] $ if $z \in \mathbb{CP}^n $
I can see how $\phi$ is continuous and passes to the quotient ( constant on the fibers of $p_2 : \mathbb{B}_{2n+2} \dot{\cup} \mathbb{CP}^n \longrightarrow \mathbb{B}_{2n+2} \cup_{p_{1}} \mathbb{CP}^n$) but i am having troubles to see that it is bijective
Any help would be appreciated.