Unclear definition of open sets in proof of Regular Lindelof space is normal

Question

I found the proof of Regular and Lindelof spaces are normal here https://math.dartmouth.edu/archive/m54x12/public_html/m54hwsolns9.pdf

and I can't understand the last part of the proof:

$$U=\bigcup_{n=1}^\infty U_n'$$ and $$V=\bigcup_{n=1}^\infty V_n'$$ are the desired open sets. Why?

Ins't enough $A\subset\bigcup_{n=1}^\infty U_n$ and $B\subset\bigcup_{n=1}^\infty V_n$?

Why to subtract the $\bigcup_{i=1}^n \overline U_i$ and $\bigcup_{i=1}^n \overline V_i$ to $ U_n$ and $ V_n$, respectively?

Answer 1

$\displaystyle\bigcup_{i=1}^{n}\overline{V_{i}}$ is closed, and so $U_{n}-\displaystyle\bigcup_{i=1}^{n}\overline{V_{i}}$ is open. If it were subtracted with $V_{i}$ instead of $\overline{V_{i}}$, the whole $U_{n}'$ is not necessarily open.