From 81a1425f873fb546988d50dc41569b04d3ce7571 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Wed, 22 Oct 2025 14:43:13 -0500 Subject: [PATCH] updates? final end of nextra. Compilation.hooks.processAssets RangeError: Invalid string length --- content/Math4201/Math4201_L23.md | 133 +++++++++++++++++++++++++++++++ content/Math4201/_meta.js | 1 + package.json | 2 +- 3 files changed, 135 insertions(+), 1 deletion(-) create mode 100644 content/Math4201/Math4201_L23.md diff --git a/content/Math4201/Math4201_L23.md b/content/Math4201/Math4201_L23.md new file mode 100644 index 0000000..33c9833 --- /dev/null +++ b/content/Math4201/Math4201_L23.md @@ -0,0 +1,133 @@ +# Math4201 Topology I (Lecture 23) + +## Connectedness of topological spaces + +### Connected space + +#### Definition of connected space + +Let $X$ be a topological space. $X$ is separated if there exist two disjoint nonempty open subsets $U,V\subset X$ such that $X=U\cup V$. + +If $X$ is not separated, then $X$ is connected. + +#### Any interval in $\mathbb{R}$ with standard topology is connected + +Let $I=[a,b]$ be an interval in $\mathbb{R}$ with standard topology. Then $I$ is connected. + +
+Proof + +By contradiction, we assume that $U,V$ give a separation of $\mathbb{R}$. In particular, $\exists a\in U, b\in V$. + +Let $a_0\coloneqq \sup\{x\in U\cap [a,b]\}$. Note that $a\in U\cap [a,b]$, so $a_0\geq a$. Since any element of $U\cap [a,b]$ is less than or equal to $b$, $a_0\leq b$. + +**Case 1**: $a_0=a$ + +Since $U$ is open, there is $\epsilon>0$ such that $[a,a+\epsilon)\subset U\cap [a,b]$. So $a_0\geq a+\epsilon>a$, which contradicts the definition of $a_0$. + +**Case 2**: $a_0=b$. + +Since $V$ is open, there is $\epsilon>0$ such that $(b-\epsilon,b]\subseteq V\cap [a,b]$. This implies that $b-\epsilon$ is also an upper bound of $U\cap [a,b]$, so $a_0\leq b-\epsilon0$ such that $(a_0-\epsilon,a_0+\epsilon)\subset U\cap [a,b]$ because $U$ is open. + +In particular, $a_0$ is greater than any element of $(a_0-\epsilon,a_0+\epsilon)$, which contradicts the definition of $a_0$. + +**Subcase II**: $a_0\in V$. + +There is an $\epsilon>0$ such that $(a_0-\epsilon,a_0+\epsilon)\subset V\cap [a,b]$ because $V$ is open. + +In particular, $a_0$ is less than any element of $(a_0-\epsilon,a_0+\epsilon)$. Since $a_0$ is an upper bound of $U\cap [a,b]$, any point $>a_0$ is not in $U\cap [a,b]$. + +So, $U\cap [a,b]\subseteq [a,a_0-\epsilon)$. This shows that $a_0-\epsilon$ is an upper bound of $U\cap [a,b]$, which contradicts the definition of $a_0$. + +
+ +_Intuitively, since both sets in $\mathbb{R}$ are open, you cannot set a clear boundary between the two sets by least upper bound argument._ + +#### Corollary as Intermediate Value Theorem + +If $f:[a,b]\to \mathbb{R}$ is continuous, and $c\in\mathbb{R}$ is such that $f(a) +Proof + +Since $[a,b]$ is connected, since $f$ is continuous, $f([a,b])$ is connected. + +By contradiction, if $c\notin f([a,b])$, then $f([a,b])$ has two points $f(a),f(b)$ and $c$ is a point between that isn't in $f([a,b])$. This contradicts the connectedness of $f([a,b])$. + +So $f(a) + +#### Definition of path-connected space + +A topological space $X$ is path-connected if for any two points $x,x'\in X$, there is a continuous map $\gamma:[0,1]\to X$ such that $\gamma(0)=x$ and $\gamma(1)=x'$. Any such continuous map is called a path from $x$ to $x'$. + +> [!NOTE] +> +> Path-connectedness is a stronger condition than connectedness. + +#### Theorem of path-connectedness and connectedness + +Any path-connected space is connected. + +
+Proof + +By contradiction, let $U,V$ be a separation of $X$. In particular, $\exists x\in U, x'\in V$. + +Since $X$ is path-connected, $\exists \gamma:[0,1]\to X$ such that $\gamma(0)=x$ and $\gamma(1)=x'$. + +Then since $\gamma$ is continuous, $\gamma^{-1}(U)$ and $\gamma^{-1}(V)$ are open in $[0,1]$ and $[0,1]=\gamma^{-1}(U)\cup \gamma^{-1}(V)$. We want to show that this gives a separation of $[0,1]$. + +Since $U\cap V=\emptyset$, $\gamma^{-1}(U)$ and $\gamma^{-1}(V)$ are disjoint. + +$U\cup V=X$ so $\gamma^{-1}(U)\cup \gamma^{-1}(V)=[0,1]$. + +Each of $\gamma^{-1}(U)$ and $\gamma^{-1}(V)$ is non-empty because $x\in U\implies 0\in \gamma^{-1}(U)$ and $x'\in V\implies 1\in \gamma^{-1}(V)$. + +This contradicts the assumption that $[0,1]$ is connected. + +
+ +
+Example of path-connected space + +A subspace $X$ of $\mathbb{R}^n$ is convex if for any two points $x,x'\in X$, the line segment connecting $x$ and $x'$ is contained in $X$. + +In particular $B_R(x)$ is convex. So $X$ is path-connected. + +--- + +Let $X=\mathbb{R}^n\setminus\{0\}$. with $n\geq 2$. Then $X$ is path-connected. (simply walk around the origin) + +
+ +#### Theorem for invariant property of connectedness + +If $f:X\to Y$ is a continuous and surjective map, and $X$ is connected, then $Y$ is connected. + +
+Proof + +Take $y,y'\in Y$, since $f$ is surjective, $\exists x,x'\in X$ such that $f(x)=y$ and $f(x')=y'$. Let $\gamma:[0,1]\to X$ be a path from $x$ to $x'$. + +Then $f\circ \gamma:[0,1]\to Y$ is a continuous map. and $f\circ \gamma(0)=y$ and $f\circ \gamma(1)=y'$. + +
+ +
+Example of connected but not path-connected space + +Let $A=\{(x,y)\in \mathbb{R}^2\mid y=\sin(1/x), x>0\}$. Then $A$ is connected, and also path-connected. + +However, take $\overline{A}=A\cup \{0\}\times [-1,1]$. Then $\overline{A}$ is not path-connected but connected. + +_Show next time_ + +
diff --git a/content/Math4201/_meta.js b/content/Math4201/_meta.js index e9d9a76..b2e8c41 100644 --- a/content/Math4201/_meta.js +++ b/content/Math4201/_meta.js @@ -26,4 +26,5 @@ export default { Math4201_L20: "Topology I (Lecture 20)", Math4201_L21: "Topology I (Lecture 21)", Math4201_L22: "Topology I (Lecture 22)", + Math4201_L23: "Topology I (Lecture 23)", } diff --git a/package.json b/package.json index a65e891..a295e38 100644 --- a/package.json +++ b/package.json @@ -1,7 +1,7 @@ { "scripts": { "dev": "next --turbopack", - "build": "cross-env VERCEL_FORCE_NO_BUILD_CACHE=1 NODE_OPTIONS='--max-old-space-size=16384' next build", + "build": "cross-env VERCEL_FORCE_NO_BUILD_CACHE=1 NODE_OPTIONS='--max-old-space-size=32768' next build", "build:test": "cross-env ANALYZE=true NODE_OPTIONS='--inspect --max-old-space-size=4096' next build", "build:analyze": "cross-env ANALYZE=true NODE_OPTIONS='--max-old-space-size=16384' next build", "postbuild": "next-sitemap && pagefind --site .next/server/app --output-path public/_pagefind && cp -r ./public/_pagefind ./out",