If I generate a topology for $\mathbb{R} \times \mathbb{R}$ by taking products of intervals $[a,b)$, how can I demonstrate that this topology has an uncountable discrete subspace?

Intuitively I think I understand how this new space will have an uncountable subspace, as it will contain a copy of $\mathbb{R}$. However, I'm having a difficult time writing out my ideas, and addressing the discreteness of the subspace.

I also know that if the subspace is discrete then any element of the subspace exists in an open subset which contains no other point of the subspace.