How to show that normality is equivalent to the intermediate open set property? (Step-1 of Urysohn Lemma in Munkres) In the process of a proof of the Urysohn Lemma, Munkres' Topology says two things that I can't prove them:
1- Let $X$ be a normal space. Let $U_p$ and $U_q$ are two open sets such that  $\bar U_p \subset U_q$. Using normality of $X$, we can always find an open set $U_r$ of $X$ such that $\ \bar U_p \subset U_r \ $ and $\ \bar U_r \subset U_q \ $. Why? I can't see any connection to the definition of normality.
2- Now let $X$ be a regular space. Considering the same things said in "1" we may not find an open set $U_r$ of $X$ such that $\ \bar U_p \subset U_r \ $ and $\ \bar U_r \subset U_q \ $. Why? I can't see any connection to the definition of regularity, as well!   

Apply the definition of normality to the disjoint closed sets $\bar{U}_p$ and $X -U_q$: there exists some open $A$ containing $\bar{U}_p$ and some open $B$ containing $X-U_q$ such that $A$ and $B$ are disjoint. Note that $\bar{A}$ and $B$ are also disjoint: every neighborhood of a point in $\bar{A}$ contains points in $A$ (which are not in $B$), so this point cannot lie in $B$.
Then take $U_r:= A$; it contains $\bar{U}_p$ by construction, and also $\bar{U}_r \subset (X-B) \subset U_q$.
