Showing that $\mathbb{P}^n \cong k^n \cup k^{n-1} \cup ... \cup k^1 \cup k^0$

Question

Given that $\mathbb{P}$ is the projective space over $k$ and $k$ is an algebraically closed field.

Answer 1

Hint: Show that $\mathbb{P}^n = k^n \cup \mathbb{P}^{n-1}$ by looking at coordinates - you don't need algebraically closed.