In the 4-dimensional Minkowski spacetime, for a given point $x=(x^0,x^1,x^2,x^3)$, its timelike future or past set is defined as, $I^{\pm}(x)= \{y=(y^0,y^1,y^2,y^3) \in \mathbb{R}^4: \eta_{\mu \nu}(y−x)^{\mu} (y−x)^{\nu}>0,\pm (y^0−x^0)>0 \}$ where $(\eta_{\mu \nu})=diag(1,−1,−1,−1)$, and one also defines the double cone sets as $\mathcal{O}(x,y)=I^+(x) \cap I^-(y)$.
I have to show that there is a countable family $ \{\mathcal{O}_n \}$ with $n \in \mathbb{N}$, of double cones in $\mathbb{R}^4$ so that every open subset $U$ of $\mathbb{R}^4$ can be written as a union of suitable elements of $\{\mathcal{O}_n\}$.
I know I have to use the second countability property, but I am not certain of how to proceed with the proof.