update notations and fix typos

pages/Math4111/Math4111_L8.md

@@ -41,7 +41,7 @@ let $B_s(p)\cap E)\backslash \{p\}={q_1,...,q_n}$
 
 Then $(B_s(p)\cap E)\backslash \{p\}=\phi$, so $p\notin E$
 
-EOP
+QED
 
 #### Theorem 2.22 De Morgan's law
 
@@ -95,7 +95,7 @@ $$
 
 So $(E^c)'\subset E^c$
 
-EOP
+QED
 
 #### Theorem 2.24
 
@@ -105,7 +105,7 @@ Proof:
 
 Suppose $\forall \alpha, G_\alpha$ is open. Let $x\in \bigcup _{\alpha} G_\alpha$. Then $\exists \alpha_0$ such that $x\in G_{\alpha_0}$. Since $G_{\alpha_0}$ is open, $\exists r>0$ such that $B_r(x)\subset G_{\alpha_0}$ Then $B_r(x)\subset G_{\alpha_0}\subset \bigcup_{\alpha} G_\alpha$
 
-EOP
+QED
 
 ##### A finite intersection of open set is open
 
@@ -117,7 +117,7 @@ Let $x\in \bigcap^n_{i=1}G_i$, then $\forall i\in \{1,..,n\}$ and $G_i$ is open,
 
 Let $r=min\{r_1,...,r_n\}$. Then $\forall i\in \{1,...,n\}$. $B_r(x)\subset B_{r_i}(x)\subset G_i$. So $B_r(x)\subset \bigcup_{i=1}^n G_i$
 
-EOP
+QED
 
 The other two can be proved by **Theorem 2.22,2.23**
 
@@ -147,4 +147,4 @@ This proves (b)
 
 So $\bar{E}^c$ is open
 
-EOP
+QED