Prove/Disprove that :
$(i)$ Every open Set in $\mathbb R^p$ can be written as the union of countable number of disjoint open Sets.
$(ii)$ Every open subset of $\mathbb R^p$ is the union of a countable collection of closed sets.
I was able to look at some similar posts asking this problem; but one seemed to be using the other and vice versa and seem convoluted.
Unfortunately, I have no idea on how to move forward. Can anyone please help me in preparing a proof for both of these problems?
Thank you for your help.