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+# Math4202 Topology II (Lecture 8)
+
+## Algebraic Topology
+
+### Path homotopy
+
+#### Recall definition of path homotopy
+
+Let $f,f':I\to X$ be a continuous maps from an interval $I=[0,1]$ to a topological space $X$.
+
+Two pathes $f$ and $f'$ are path homotopic if
+
+- there exists a continuous map $F:I\times [0,1]\to X$ such that $F(i,0)=f(i)$ and $F(i,1)=f'(i)$ for all $i\in I$.
+- $F(s,0)=f(0)$ and $F(s,1)=f(1)$, $\forall s\in I$.$F(s,0)=f(0)$ and $F(s,1)=f(1)$, $\forall s\in I$
+
+#### Lemma: Homotopy defines an equivalence relation
+
+The $\simeq$, $\simeq_p$ are both equivalence relations.
+
+<details>
+<summary>Proof</summary>
+
+**Reflexive**:
+
+$f:I\to X$, $F:I\times I\to X$, $F(s,t)=f(s)$.
+
+$F$ is a homotopy between $f$ and $f$ itself.
+
+**Symmetric**:
+
+Suppose $f,g:I\to X$,
+
+$F:I\times I\to X$ is a homotopy between $f$ and $g$.
+
+Let $H: I\times I\to X$ be a homotopy between $g$ and $f$ defined as follows:
+
+$H(s,t)=F(s,1-t)$.
+
+$H(s,0)=F(s,1)=g(s)$, $H(s,1)=F(s,0)=f(s)$.
+
+Therefore $H$ is a homotopy between $g$ and $f$.
+
+**Transitive**:
+
+Suppose we have $f\simeq_p g$ with homotopy $F_1$, and $g\simeq_p h$ with homotopy $F_2$.
+
+Then we can glue the two homotopies together to get a homotopy $F$ between $f$ and $h$ using pasting lemma.
+
+$F(s,t)=(F_1*F_2)(s,t)\coloneqq\begin{cases}
+F_1(s,2t), & t\in [0,\frac{1}{2}]\\
+F_2(s,2t-1), & t\in [\frac{1}{2},1]
+\end{cases}$
+
+Therefore $f\simeq_p h$ with homotopy $F$.
+
+</details>
+
+> [!NOTE]
+>
+> We use $[x]$ to denote the equivalence class of $x$.
+
+<details>
+<summary>Example of equivalence classes in path homotopy</summary>
+
+Let $X=\{pt\}$, $\operatorname{Path}(X)=\{\text{constant map}\}$.$\operatorname{Path}/_{\simeq_p}(X)=\{[\text{constant map}]\}$.
+
+---
+
+$X=\{p,q\}$ with discrete topology, $\operatorname{Path}(X)=\{f_{p},f_{q}\}$.$\operatorname{Path}/_{\simeq_p}(X)=\{[f_{p}], [f_{q}]\}$
+
+This applied to all discrete topological spaces.
+
+---
+
+Let $X=\mathbb{R}$ with standard topology.
+
+$\operatorname{Path}(X)=\{f:[0,1]\to \mathbb{R}\in C^0\}$
+
+Let $f_1,f_2:[0,1]\to \mathbb{R}$ where $f_1(0)=f_2(0)$, $f_1(1)=f_2(1)$.
+
+Then we can construct a homotopy between $f_1$ and $f_2$.
+
+$F:[0,1]\times [0,1]\to \mathbb{R}$, $F(s,t)=(1-t)f_1(s)+tf_2(s)$ is a homotopy between $f_1$ and $f_2$.
+
+$\operatorname{Path}/_{\simeq_p}(X)=\{(x_1,x_1)|x_1,x_2\in \mathbb{R}\}$
+
+This applies to any convex space $V$ in $\mathbb{R}^n$.
+</details>