[Math] In $\mathbb R^p$:Every open subset is the union of a countable collection of closed sets & every open set is the countable union of disjoint open sets

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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.

Best Answer

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  1. Every open set is union of balls with rational radius and rational center.
  2. Every open ball is a countable union of closed balls.

This gives (ii). For (i), given two points in your open set, say that they are equivalent iff there is a continuous path between them, completely contained in the open set. Argue that this is indeed an equivalence relation, and that its components (equivalence classes) are open. Now use that $\mathbb Q^n$ is dense in $\mathbb R^n$, so there can be no more than countably many equivalence classes.

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