Verifying if it is a legitimate basis for a topology

Question

I have been given the task of showing that $\mathcal{B}:=\{[a,b)\vert -\infty<a,b<\infty\}$ is a basis for a topology on $\mathbb{R}$.
My problem with it is that there is not a finite union of sets in $\mathcal{B}$ which gives us $\mathbb{R}$, so how can this be a basis for a topology on $\mathbb{R}$?

Answer 1

Because $\Bbb R=\bigcup_{n\in\Bbb Z}[n,n+1)$. Yes, $\Bbb R$ is not a finite union of elements of $\mathcal B$. So what? All it has to be is an union of elements of $\mathcal B$.