update notations and fix typos

pages/Math4111/Math4111_L7.md

@@ -80,12 +80,16 @@ Let $(X,d)$ be  a metric space, $\forall p\in X,\forall r>0$, $B_r(p)$ is an ope
 
 *every ball is an open set*
 
-Proof: Let $q\in B_r(p)$.
+Proof:
+
+Let $q\in B_r(p)$.
 
 Let $h=r-d(p,q)$.
 
 Since $q\in B_r(p),h>0$. We claim that $B_h(q)$. Then $d(q,s)<h$, so $d(p,s)\leq d(p,q)+d(q,s)<d(p,q)+h=r$. (using triangle inequality) So $S\in B_r(p)$.
 
+QED
+
 ### Closed sets
 
 1. $E\subset X,p\in X$. We say $p$ is a limit point of $E$ if $\forall r>0, (B_r(p)\cap E)\backslash {p}\neq \phi$.
@@ -94,7 +98,9 @@ Since $q\in B_r(p),h>0$. We claim that $B_h(q)$. Then $d(q,s)<h$, so $d(p,s)\leq
 
 2. $E$ is closed if $E'\subset E$
 
-Example: $X=\mathbb{R}^2$, $E=[0,1)\times [0,1)$.
+Example:
+
+$X=\mathbb{R}^2$, $E=[0,1)\times [0,1)$.
 
 $(1,1)$ is a limit point.