How do i prove that every open set in $\mathbb{R}^2$ is a union of at most countable open rectangles?
Let $E$ be open in $\mathbb R^k$. For each $x\in E$, there is an open ball centred at $x$ that is contained in $E$. Now one can construct an open $k$-cell containing $x$ and contained in this ball of the form $(a_1,b_1)\times ...\times (a_k,b_k)$ where $a_1,b_1,...,a_k,b_k$ are all rational, and then $E$ would be the union of these cells. This union is at most countable (after discarding identical cells) because $\{(a_1,b_1)\times ...\times (a_k,b_k):a_1,b_1,...,a_k,b_k\in\mathbb Q\}$ is countable.