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content/Math4201/Math4201_L2.md

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+# Math4201 Lecture 2
+
+## Topology
+
+Distance in $\mathbb{R}$, or more generally in $\mathbb{R}^n$. ([metric space](https://notenextra.trance-0.com/Math4111/Math4111_L9#metric-space))
+
+Intervals in $\mathbb{R}$, or more generally open balls in $\mathbb{R}^n$. (topological space)
+
+### Topological Spaces
+
+#### Definition of Topological Space
+
+A topological space is a pair of set $X$ and a collection of subsets of $X$, denoted by $\mathcal{T}$ (imitates the set of "open sets" in $X$), satisfying the following axioms:
+
+1. $\emptyset \in \mathcal{T}$ and $X \in \mathcal{T}$
+2. $\mathcal{T}$ is closed with respect to arbitrary unions. This means, for any collection of open sets $\{U_\alpha\}_{\alpha \in I}$, we have $\bigcup_{\alpha \in I} U_\alpha \in \mathcal{T}$
+3. $\mathcal{T}$ is closed with respect to finite intersections. This means, for any finite collection of open sets $\{U_1, U_2, \ldots, U_n\}$, we have $\bigcap_{i=1}^n U_i \in \mathcal{T}$
+
+The elements of $\mathcal{T}$ are called open sets.
+
+The topological space is denoted by $(X, \mathcal{T})$.
+
+#### Examples
+
+Trivial topology: Let $X$ be arbitrary. The trivial topology is $\mathcal{T}_0 = \{\emptyset, X\}$
+
+Discrete topology: Let $X$ be arbitrary. The discrete topology is $\mathcal{T}_1 = \mathcal{P}(X)=\{U \subseteq X\}$
+
+Understanding all possible topologies on a set.
+
+Let's say $X=\{a,b\}$
+
+The trivial topology is $\mathcal{T}_0 = \{\emptyset, \{a,b\}\}$
+
+The discrete topology is $\mathcal{T}_1 = \{\emptyset, \{a\}, \{b\}, \{a,b\}\}$
+
+Other topologies:
+
+$\mathcal{T}_2 = \{\emptyset, \{a\}, \{a,b\}\}$
+
+$\mathcal{T}_3 = \{\emptyset, \{b\}, \{a,b\}\}$
+
+Non-examples:
+
+Let $X=\{a,b,c\}$
+
+The set $\mathcal{T}_1=\{\emptyset, \{a\}, \{b\}, \{a,b,c\}\}$ is not a topology because it is not closed under union $\{a\} \cup \{b\} = \{a,b\} \notin \mathcal{T}_1$
+
+#### Definition of Complement finite topology
+
+Let $X$ be arbitrary. The complement finite topology is $\mathcal{T}\coloneqq \{U\subseteq X|X\setminus U \text{ is finite}\}\cup \{\emptyset\}$
+
+The topology is valid because:
+
+<details>
+<summary>Proof</summary>
+
+1. $\emptyset \in \mathcal{T}$ because $X\setminus \emptyset = X$ is finite.
+2. Let $\{U_\alpha\}_{\alpha \in I}$ be an arbitrary collection such that $X\setminus U_\alpha$ is finite for each $\alpha \in I$. 
+   
+   Without loss of generality, we can assume that $U_\alpha \neq \emptyset$ for each $\alpha \in I$, since the union of arbitrary set with $\emptyset$ is the set itself. 
+   
+   If all of them are empty, then the union is empty, which complement is $X\subset \mathcal{T}$.
+   
+   Otherwise, 
+   
+   $$
+   X\setminus \bigcup_{\alpha \in I} U_\alpha = \bigcap_{\alpha \in I} (X\setminus U_\alpha)
+   $$
+   is finite because each $X\setminus U_\alpha$ is finite. Therefore, $\bigcup_{\alpha \in I} U_\alpha \in \mathcal{T}$.
+3. Let $\{U_1, U_2, \ldots, U_n\}$ be a finite collection such that $X\setminus U_i$ is finite for each $i=1,2,\ldots,n$. 
+
+   Without loss of generality, we can assume that $U_i \neq \emptyset$ for each $i=1,2,\ldots,n$, since the intersection of arbitrary set with $\emptyset$ is $\emptyset$.
+   
+   If all of them are empty, then the intersection is $X\subset \mathcal{T}$.
+   
+   Otherwise,
+   
+   $$
+   X\setminus \bigcap_{i=1}^n U_i = \bigcup_{i=1}^n (X\setminus U_i)
+   $$
+   is finite because each $X\setminus U_i$ is finite. Therefore, $\bigcap_{i=1}^n U_i \in \mathcal{T}$.
+</details>
+
+Another non-example is $\mathcal{T} = \{U\subseteq X|U \text{ is finite}\}\cup \{X\}$
+
+The topology is invalid because an arbitrary union of points is not a finite set.
+
+Consider $X=\mathbb{Z}$ but take $U_1=\{1\}, U_2=\{2\}, U_3=\{3\}, \ldots$
+
+Then $\bigcup_{i=1}^\infty U_i = \mathbb{Z}^+$ is not a finite set and is not $\{X\}$.
+
+> [!NOTE]
+>
+> If $X$ is finite, then finite complement topology is the same as the discrete topology.
+
+#### Definition of Topology basis
+
+For a set $X$, a topology basis, denoted by $\mathcal{B}$, is a collection of subsets of $X$, such that the following properties are satisfied:
+
+1. For any $x \in X$, there exists a $B \in \mathcal{B}$ such that $x \in B$
+2. If $B_1, B_2 \in \mathcal{B}$ and $x \in B_1 \cap B_2$, then there exists a $B_3 \in \mathcal{B}$ such that $x \in B_3 \subseteq B_1 \cap B_2$

content/Math4201/_meta.js

@@ -4,4 +4,5 @@ export default {
         type: 'separator'
     },
     Math4201_L1: "Topology I (Lecture 1)",
+    Math4201_L2: "Topology I (Lecture 2)",
 }

content/index.md

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 # Welcome to NoteNextra
 
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+> 
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