updates

content/Math4201/Math4201_L1.md

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-# Math4201 Lecture 1
+# Math4201 Topology I (Lecture 1)
 
 Going through the syllabus.

content/Math4201/Math4201_L10.md

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-# Math4201 Lecture 10
+# Math4201 Topology I (Lecture 10)
 
 ## Continuity
 
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 Let $X,Y$ be topological spaces and $f:X\to Y$. For any $x\in X$ and any open neighborhood $V$ of $f(x)$ in $Y$, $f^{-1}(V)$ contains an open neighborhood of $x$ in $X$.
 
-
 #### Lemma for continuous functions
 
 Let $f:X\to Y$ be a function, then:

content/Math4201/Math4201_L11.md

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+# Math4201 Topology I (Lecture 11)
+
+> [!NOTE]
+>
+> Q: Let $f:X\to Y$ be a continuous bijection. Is it true that $f^{-1}$ is continuous?
+>
+> A: No. Consider $X=[0,2\pi)$ and $Y=\mathbb{S}^1$ with standard topology in $\mathbb{R}^2$.
+>
+> Let $f\coloneq \theta\in [0,2\pi)\to (\cos \theta, \sin \theta)\in \mathbb{S}^1$ is a continuous bijection. ($\forall f^{-1}(V)$ is open in $X$)
+>
+> But $f^{-1}$ is not continuous, consider the open set in $X, U=[0,\pi)$. Then $f^{-1}(U)=[0,\pi)$ is not open in $Y$.
+
+## Continuous functions
+
+### Constructing continuous functions
+
+#### Theorem composition of continuous functions is continuous
+
+Let $X,Y,Z$ be topological spaces, $f:X\to Y$ is continuous, and $g:Y\to Z$ is continuous. Then $f\circ g:X\to Z$ is continuous.
+
+<details>
+<summary>Proof</summary>
+
+Let $U\subseteq Z$ be open. Then $g^{-1}(U)$ is open in $Y$. Since $f$ is continuous, $f^{-1}(g^{-1}(U))$ is open in $X$.
+
+</details>
+
+#### Pasting lemma
+
+Let $X$ be a topological space and $X=Z_1\cup Z_2$ with $Z_1,Z_2$ closed in $X$ equipped with the subspace topology. (may be not disjoint)
+
+Let $g_1:Z_1\to Y$ and $g_2:Z_2\to Y$ be two continuous maps and $\forall x\in Z_1\cap Z_2$, $g_1(x)=g_2(x)$.
+
+Define $f:X\to Y$ by $f(x)\begin{cases}g_1(x), & x\in Z_1 \\ g_2(x), & x\in Z_2\end{cases}$ is continuous.
+
+<details>
+<summary>Proof</summary>
+
+Let $U\subseteq Y$ be open. Then $f^{-1}(U)=g_1^{-1}(U)\cup g_2^{-1}(U)$.
+
+$g_1^{-1}(U)$ and $g_2^{-1}(U)$ are open in $Z_1$ and $Z_2$ respectively.
+
+> It's a bit annoying to show that $g_1^{-1}(U)$ and $g_2^{-1}(U)$ are open in $X$.
+
+Different way. Consider the definition of continuous functions using closed sets.
+
+If $W\subseteq X$ is closed, then $W=Z_1\cap Z_2$ is closed in $X$.
+
+So $f^{-1}(W)=g_1^{-1}(W)\cup g_2^{-1}(W)$ is closed in $Z_1$ and $Z_2$ respectively.
+
+Note that $Z_1$ and $Z_2$ are closed in $X$, so $g_1^{-1}(W)$ and $g_2^{-1}(W)$ are closed in $X$. [closed in closed subspace lemma](https://notenextra.trance-0.com/Math4201/Math4201_L7#lemma-of-closed-in-closed-subspace)
+
+So $f^{-1}(W)$ is closed in $X$.
+
+</details>
+
+Let $X$ be a topological space and $X=U_1\cup U_2$ with $U_1,U_2$ open in $X$ equipped with the subspace topology.
+
+With $g_1:U_1\to Y$ and $g_2:U_2\to Y$ be two continuous maps and $\forall x\in U_1\cap U_2$, $g_1(x)=g_2(x)$.
+
+Then $f:X\to Y$ by $f(x)\begin{cases}g_1(x), & x\in U_1 \\ g_2(x), & x\in U_2\end{cases}$ is continuous.
+
+<details>
+<summary>Proof</summary>
+
+Let $U\subseteq Y$ be open. Then $f^{-1}(U)=g_1^{-1}(U)\cup g_2^{-1}(U)$.
+
+$g_1^{-1}(U)$ and $g_2^{-1}(U)$ are open in $U_1$ and $U_2$ respectively.
+
+Apply the [open in open subspace lemma](https://notenextra.trance-0.com/Math4201/Math4201_L6#lemma-of-open-set-in-subspace-topology)
+
+So $f^{-1}(U)$ is open in $X$.
+
+</details>
+
+The open set version holds more generally.
+
+Let $X$ be a topological space and $X=\bigcup_{\alpha\in I} U_\alpha$ with $U_\alpha$ open in $X$ equipped with the subspace topology.
+
+Let $g_\alpha:U_\alpha\to Y$ be two continuous maps and $\forall x\in U_\alpha\cap U_\beta$, $g_\alpha(x)=g_\beta(x)$.
+
+Then $f:X\to Y$ by $f(x)=g_\alpha(x), \text{if } x\in U_\alpha$ is continuous.
+
+#### Continuous functions on different codomains
+
+Let $f:X\to Y$ and $g:X\to Z$ be two continuous maps of topological spaces.
+
+Let $H:X\to Y\times Z$, where $Y\times Z$ is equipped with the product topology, be defined by $H(x)=(f(x),g(x))$. Then $H$ is continuous.
+
+> A stronger version of this theorem is that $f:X\to Y$ and $g:X\to Z$ are continuous maps of topological spaces if and only if $H:X\to Y\times Z$ is continuous.
+
+<details>
+<summary>Proof</summary>
+
+It is sufficient to check the basis elements of the topology on $Y\times Z$.
+
+The basis for the topology on $Y\times Z$ is $U\times V\subseteq Y\times Z$, where $U\subseteq Y$ and $V\subseteq Z$ are open. This form a basis for the topology on $Y\times Z$.
+
+We only need to show that $H^{-1}(U\times V)$ is open in $X$.
+
+Let $H^{-1}(U\times V)=\{x\in X | (f(x),g(x))\in U\times V\}$.
+
+So $H^{-1}(U\times V)=f^{-1}(U)\cap g^{-1}(V)$.
+
+Since $f$ and $g$ are continuous, $f^{-1}(U)$ and $g^{-1}(V)$ are open in $X$.
+
+So $H^{-1}(U\times V)$ is open in $X$.
+
+</details>
+
+Exercise: Prove the stronger version of the theorem,
+
+If $H:X\to Y\times Z$ is continuous, then $f:X\to Y$ and $g:X\to Z$ are continuous.

content/Math4201/Math4201_L2.md

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-# Math4201 Lecture 2
+# Math4201 Topology I (Lecture 2)
 
 ## Topology
 

content/Math4201/Math4201_L3.md

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-# Math4201 Lecture 3
+# Math4201 Topology I (Lecture 3)
 
 ## Recall form last lecture
 

content/Math4201/Math4201_L4.md

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-# Math4201 Lecture 4
+# Math4201 Topology I (Lecture 4)
 
 ## Recall from last lecture
 

content/Math4201/Math4201_L5.md

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-# Math4201 Lecture 5 Bonus
+# Math4201 Topology I (Lecture 5 Bonus)
 
 ## Comparison of two types of topologies
 

content/Math4201/Math4201_L6.md

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-# Math4201 Lecture 6
+# Math4201 Topology I (Lecture 6)
 
 ## Product topology
 

content/Math4201/Math4201_L7.md

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-# Math 4201 Lecture 7
+# Math 4201 Topology I (Lecture 7)
 
 ## Review from last lecture
 

content/Math4201/Math4201_L8.md

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-# Math4201 Lecture 8
+# Math4201 Topology I (Lecture 8)
 
 Recall from real analysis, a set is closed if and only if it has limit points.
 

content/Math4201/Math4201_L9.md

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-# Math4201 Lecture 9
+# Math4201 Topology I (Lecture 9)
 
 ## Convergence of sequences
 

content/Math4201/_meta.js

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     Math4201_L8: "Topology I (Lecture 8)",
     Math4201_L9: "Topology I (Lecture 9)",
     Math4201_L10: "Topology I (Lecture 10)",
+    Math4201_L11: "Topology I (Lecture 11)",
 }

content/Math4201/index.md

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-# Math4201
+# Math4201 Course Description
 
 E-mail: adaemi@wustl.edu