real analysis - Which topology do we need to assume to get this result: Every open set $V$ in the plane is a countable union of rectangles

I am self-studying real and complex analysis by W.Rudin and in a proof it is stated, that every open set $V$ in the plane is a countable union of rectangles $R$.

I have the following definitions at hand and am trying to proof this result.

1) A set $S$ is open iff $S$ is a member of the topology on X.

2) The topology of $\mathbb{R}$ is the set of all unions of segments in $\mathbb{R}$, e.g. sets of the form $(a,b)$.

3) The topology of $\mathbb{R^2}$ is the set of all unions of open circular discs.

4) A rectangle is defined as following $R=I_1\times I_2$, where $I_1,I_2$ are segments in $\mathbb{R}$

I can prove the result by assuming the euclidian metric and the topology resulting from the usual definition of open balls: $\{x \in \mathbb{R}: |a-x|

Help would be very much appreciated.