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$\mathbb{CP}^1$ is the set of all one dimensional subspaces of $\mathbb{C}^2$. Let $(z,w)\in \mathbb{C}^2$ be non zero; its span is a point in $\mathbb{CP}^1$. Let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and $[z:w]=[\lambda z:\lambda w],\lambda\in\mathbb{C}^{*}$ is a point in $\mathbb{CP}^1$, the map is $\phi_0:U_0\rightarrow\mathbb{C}$ defined by $\phi_0([z:w])=w/z$ the map $\phi:U_1\rightarrow\mathbb{C}$ defined by $\phi_1([z:w])=z/w$ could any one tell me why the inverses of these maps are continuous?

$\bullet$ What is the Homeomorphism between $\mathbb{C}P^1$ and $\mathbb{C}\cup\{\infty\}$?

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    Notice that this definition is still ambiguous because you need to define a topology on $(\mathbb{C}^2)^*$ to be able to define a quotient topology on $\mathbb{CP}^1$. Although, it would most probably be the usual metric topology defined by the metric $d : (\mathbb{C}^2)^* \to \mathbb{C}$ where $d(z_1,z_2) := |z_1 - z_2|.$2012-09-04

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