In the projective space $\mathbb{P}^n(\mathbb{K})$ I can consider the hyerplane $H_{0}=\{x_{0}=0\}$ and the set $U_{0}=\mathbb{P}^n(\mathbb{K})-H_{0}$. Clearly $\mathbb{P}^n(\mathbb{K})=U_{0}\cup H_{0}$. The functions
- $j_{0}:\mathbb{K}^n\to U_{0}$ defined by $j_{0}(x_{1},...,x_{n})=[1,x_{1},...,x_{n}]$,
- $i_{0}:H_{0}\to \mathbb{P}^{n-1}(\mathbb{K})$ defined by $i_{0}([0,x_{1},...,x_{n}])=[x_{1},...,x_{n}]$.
Are obviously both bijections. My question is: are them also homeomorphisms? Why?