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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

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    Look at $g_{1}(z_2,\dots,z_n)=[1,z_2,\dots,z_n]$. Observe we can write it as $g_{1}(z)=[1,\mathrm{id}(z)]$. You can use the definition of continuity from analysis, using $\varepsilon-\delta$ proof taking $\delta=\varepsilon$ you have it immediately...2012-11-23

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Firstly, you need to prove that $\mathbb{C}P^n$ is Hausdorff. It is proved here.

Let $X = \mathbb{C}^{n+1} - \{0\}$. Let $\pi \colon X \rightarrow \mathbb{C}P^n$ be the canonical map. Let $h_0\colon \mathbb{C}^n \rightarrow X$ be the map defined by $h_0(z_1,\dots,z_n) = (1,z_1,\dots,z_n)$. Clearly $h_0$ is continuous. Since $g_0 = \pi h_0$ and $\pi$ is continuous , $g_0$ is continuous. Similarly $g_i$ is continuous for $i \neq 0$.

For the sake of completeness, I will prove that $f_i$ is continous. Since $f_0\pi(z_0,z_1,\dots,z_n) = (z_1/z_0,\dots, z_n/z_0)$, $f_0\pi\colon \pi^{-1}(U_0) \rightarrow \mathbb{C}^n$ is continous. Let $V$ be an open subset of $\mathbb{C}^n$. Since $(f_0\pi)^{-1}(V) = \pi^{-1}(f_0^{-1}(V))$ is open in $\pi^{-1}(U_0)$, it is open in $X$. Hence $f_0^{-1}(V)$ is open. Hence $f_0$ is continous. Similarly $f_i$ is continuous for $i \neq 0$.

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    But the link you provided for the proof of Hausdorffness is valid for real projective spaces. How can we do it for complex projective spaces?2018-08-07