Exercise 2.5.4 (Urysohn's Lemma II & Tietze Extension Theorem II)

Proof. Suppose that $X$ is a topological space on which Urysohn’s Lemma holds, i.e., for any two disjoint closed subsets $E$ and $F$ of $X$, there exists a continuous function $f$ from $X$ to the unit interval $[0,1]$ such that $f=0$ on $E$ and $f=1$ on F. Then the sets

must be open as the inverses of open subsets of $[0,1]$ under $f$. Furthermore, these sets contain $E$ and $F$ respectively by definition. In other words, $X$ is normal. □

Proof. Suppose that $X$ is a topological space on which any bounded continuous real-valued function $f$ on a closed subset $Y$ can be extended to a bounded continuous real-valued function $h$ on whole $X$. Let $E$ and $F$ be arbitrary disjoint closed subsets of $X$. Then $1_E$¹ is a bounded continuous real-valued function on $E\cup F$ (The inverse image of any closed set in $[0,1]$ either contains $0$ or not; in both cases, the inverse is either $E$ or $F$, i.e, it is closed by Exercise 3.2.). Let $f:X\to [0,1]$ be a continuous extension of $1_E$ to $X$. In a similar manner, the sets

are disjoint open subsets of $X$ containing $E$ and $F$ respectively. In other words, $X$ is normal. □