How to show this is an embedding from ${\mathbb{R^2}\to \mathbb{P}_2(\mathbb{R})}$?

Question

I know intuitively that the mapping given by $$ p : \mathbb{R}^2 \to \{[(x,y,1)]\ |\ x,y \in \mathbb{R}\}\subset \mathbb{P}_2(\mathbb{R}), p((x,y)) = [(x,y,1)] $$ will be an embedding from ${\mathbb{R}^2}$ to ${\{[(x,y,1)]\ |\ x,y \in \mathbb{R}\}\subset \mathbb{P}_2(\mathbb{R})}$. But how would I show it formally? It's clearly bijective, but I need to show continuity of $p$ and it's inverse, which involves looking at the open subsets of ${\{[(x,y,1)]\ |\ x,y \in \mathbb{R}\}}$, for which I am a bit stuck