I'm struggling to prove that $\mathbb CP^n$ is 2n-manifold.
We can defined the $\mathbb CP^n$ as the equivalence relation $(z_1,z_1,...,z_{n+1})\sim(w_1,w_1,...,w_{n+1})$ iff $z_i=\lambda w_i$, $i=1,2,...,n+1$.
In order to prove that $\mathbb CP^n$ is a $2n$-manifold, we need to define a function $f_i:U_i\to \mathbb C^n$ defined by $f_i([z_1,...,z_{n+1}])=\left(\frac{z_1}{z_i},...,\frac{z_{i-1}}{z_i},\frac{z_{i+1}}{z_i},..., \frac{z_{n+1}}{z_i}\right)$, where each $U_i$ is defined as $U_i = \{[z_0,z_1,...,z_n];z_i\neq 0\}$.
If we prove that this function is an homeomorphism, we're done.
It's easy to prove that each $f_i$ is well-defined, continuous and have this inverse $g_i:\mathbb C^n\to U_i$, defined by $g_i(z_1,...,z_n)=[z_1,...,z_{i-1},1,z_i,...,z_n]$
In order to prove that $\mathbb CP^n$ is a $2n$-manifold, it miss just the continuity of $g$, I need help in this part.
Thanks