Open subset of $\mathbb {CP}^n$?

Question

As picture below, how to show the $U_i$ is open subset of $\mathbb {CP}^n$ ?

Answer 1

Consider the quotient map $f: \mathbb{C}^{n+1}\rightarrow \mathbb{CP}^n$ by $f(z)=[z]$.

Clearly, $V_i = f^{-1}(U_i)$ is the set of all points in $\mathbb{C}^{n+1}$ with $i$-th coordinate is not equal to $0$. Then what is the complement of $V_i$? It is a hyperplane in $\mathbb{C}^{n+1}$ which is closed.

For example, for $n=1$, $(V_0)^c$ is the $x$-axis and $(V_1)^c$ is the $y$-axis whereas for $n=2$, $(V_0)^c$ is the $yz$-plane.

Answer 2

The space is irreducible, and so having $U_i$ as a dense subset, means that $U_i$ is open.