updates?

content/Math4201/Exam_reviews/Math4201_E3.md

@@ -635,6 +635,7 @@ A topological space $(X,\mathcal{T})$ is normal if for any disjoint closed sets
 Some corollaries:
 
 1. $X$ is normal if and only if given a closed set $A\subseteq X$, there is open neighborhood $V$ of $A$ such that $\overline{V}\subseteq U$.
+2. Every compact Hausdorff spaces is normal.
 
 > [!CAUTION]
 >
@@ -646,4 +647,12 @@ Let $X$ be a regular space with countable basis, then $X$ is normal.
 
 *Prove by taking disjoint open neighborhoods by countable cover.*
 
+### Urysohn Lemma
+
+Let $X$ be a normal space, $A,B$ be two closed disjoint set in $X$, then there exists continuous function: $f:X\to[0,1]$ such that $f(A)=\{0\}$ and $f(B)=\{1\}$.
+
+#### Urysohn metrization theorem
+
+If $X$ is normal (regular and second countable) topological space, then $X$ is metrizable.
+