update formats and lecture notes

content/Math4111/Math4111_L6.md

@@ -145,7 +145,8 @@ Since $b$ is fixed, so this is in 1-1 correspondence with $A$, so it's countable
 
 Let $A$ be the set of all sequences for 0s and 1s. Then $A$ is uncountable.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Let $E\subset A$ be a countable subset. We'll show $A\backslash E\neq \phi$ (i.e.$\exists t\in A$ such that $t\notin E$)
 
@@ -155,4 +156,4 @@ Then we define a new sequence $t$ which differs from $S_1$'s first bit and $S_2$
 
 This is called Cantor's diagonal argument.
 
-QED
+</details>

content/Math4111/Math4111_L8.md

@@ -24,7 +24,8 @@ It should be empty. Proof any point cannot be in two balls at the same time. (By
 
 $p\in E'\implies \forall r>0,B_r(p)\cap E$ is infinite.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 We will prove the contrapositive.
 
@@ -41,7 +42,7 @@ let $B_s(p)\cap E)\backslash \{p\}={q_1,...,q_n}$
 
 Then $(B_s(p)\cap E)\backslash \{p\}=\phi$, so $p\notin E$
 
-QED
+</details>
 
 #### Theorem 2.22 De Morgan's law
 
@@ -68,7 +69,8 @@ $E$ is open $\iff$ $E^c$ is closed.
 >$\phi$, $\R$ is both open and closed. "clopen set"  
 >$[0,1)$ is not open and not closed. bad...
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 $\impliedby$ Suppose $E^c$ is closed. Let $x\in E$, so $x\notin E^c$
 
@@ -95,21 +97,23 @@ $$
 
 So $(E^c)'\subset E^c$
 
-QED
+</details>
 
 #### Theorem 2.24
 
 ##### An arbitrary union of open sets is open
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 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 $B_r(x)\subset G_{\alpha_0}\subset \bigcup_{\alpha} G_\alpha$
 
-QED
+</details>
 
 ##### A finite intersection of open set is open
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Suppose $\forall i\in \{1,...,n\}$, $G_i$ is open.
 
@@ -117,7 +121,7 @@ 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 $B_r(x)\subset \bigcup_{i=1}^n G_i$
 
-QED
+</details>
 
 The other two can be proved by **Theorem 2.22,2.23**
 
@@ -131,7 +135,8 @@ Remark: Using the definition of $E'$, we have, $\bar{E}=\{p\in X,\forall r>0,B_r
 
 $\bar {E}$ is closed.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 We will show $\bar{E}^c$ is open.
 
@@ -147,4 +152,4 @@ This proves (b)
 
 So $\bar{E}^c$ is open
 
-QED
+</details>