From f9c5889564e9a6de45a95505851d2953ab2f53d3 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 9 Oct 2025 21:37:22 -0500 Subject: [PATCH] fix typo and formatting errors --- content/Math4201/Exam_reviews/Math4201_E1.md | 94 +++++++++++++++++++- content/Math4201/Math4201_L16.md | 35 ++++---- content/Math4201/Math4201_L18.md | 2 +- content/Math4201/Math4201_L2.md | 12 ++- content/Math4201/Math4201_L3.md | 8 +- content/Math4201/Math4201_L4.md | 2 +- content/Math4201/Math4201_L5.md | 2 +- 7 files changed, 130 insertions(+), 25 deletions(-) diff --git a/content/Math4201/Exam_reviews/Math4201_E1.md b/content/Math4201/Exam_reviews/Math4201_E1.md index cbc4909..dcfedc6 100644 --- a/content/Math4201/Exam_reviews/Math4201_E1.md +++ b/content/Math4201/Exam_reviews/Math4201_E1.md @@ -1 +1,93 @@ -# Math 4201 Exam 1 review \ No newline at end of file +# Math 4201 Exam 1 review + +> [!NOTE] +> +> This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it. + +The exam will have 5 problems, roughly covering the following types of questions: + +- Define concepts from class (e.g. what is the definition of the interior of a set?) +- Give an example of a space/map which satisfies/does not satisfy a certain property (e.g. give an example of a map that is not continuous.) +- Proofs from the lectures +- Homework problems +- A new problem at the same level of difficulty as homework problems + +## Topological space + +### Basic definitions + +#### Definition for 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 $\bigcap_{i=1}^n U_i \in \mathcal{T}$ + +#### Definition of open set + +$U\subseteq X$ is an open set if $U\in \mathcal{T}$ + +#### Definition of closed set + +$Z\subseteq X$ is a closed set if $X\setminus Z\in \mathcal{T}$ + +### Basis + +#### Definition of topological 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 $x \in B$ (basis covers the whole space) +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 $x \in B_3 \subseteq B_1 \cap B_2$ (every non-empty intersection of basis elements are also covered by a basis element) + +#### Definition of topology generated by basis + +Let $\mathcal{B}$ be a basis for a topology on a set $X$. Then the topology generated by $\mathcal{B}$ is defined by the set as follows: + +$$ +\mathcal{T}_{\mathcal{B}} \coloneqq \{ U \subseteq X \mid \forall x\in U, \exists B\in \mathcal{B} \text{ such that } x\in B\subseteq U \} +$$ + +> This is basically a closure of $\mathcal{B}$ under arbitrary unions and finite intersections + +#### Lemma of topology generated by basis + +$U\in \mathcal{T}_{\mathcal{B}}\iff \exists \{B_\alpha\}_{\alpha \in I}\subseteq \mathcal{B}$ such that $U=\bigcup_{\alpha \in I} B_\alpha$ + +#### Definition of basis generated from a topology + +Let $(X, \mathcal{T})$ be a topological space. Then the basis generated from a topology is $\mathcal{C}\subseteq \mathcal{B}$ such that $\forall U\in \mathcal{T}$, $\forall x\in U$, $\exists B\in \mathcal{C}$ such that $x\in B\subseteq U$. + +#### Definition of subbasis of topology + +A subbasis of a topology is a collection $\mathcal{S}\subseteq \mathcal{T}$ such that $\bigcup_{U\in \mathcal{S}} U=X$. + +#### Definition of topology generated by subbasis + +Let $\mathcal{S}\subseteq \mathcal{T}$ be a subbasis of a topology on $X$, then the basis generated by such subbasis is the closure of finite intersection of $\mathcal{S}$ + +$$ +\mathcal{B}_{\mathcal{S}} \coloneqq \{B\mid B\text{ is the intersection of a finite number of elements of }\mathcal{S}\} +$$ + +Then the topology generated by $\mathcal{B}_{\mathcal{S}}$ is the subbasis topology denoted by $\mathcal{T}_{\mathcal{S}}$. + +Note that all open set with respect to $\mathcal{T}_{\mathcal{S}}$ can be written as a union of finitely intersections of elements of $\mathcal{S}$ + +### Comparing topologies + +#### Definition of finer and coarser topology + +Let $(X,\mathcal{T})$ and $(X,\mathcal{T}')$ be topological spaces. Then $\mathcal{T}$ is finer than $\mathcal{T}'$ if $\mathcal{T}'\subseteq \mathcal{T}$. $\mathcal{T}$ is coarser than $\mathcal{T}'$ if $\mathcal{T}\subseteq \mathcal{T}'$. + +#### Lemma of comparing basis + +Let $(X,\mathcal{T})$ and $(X,\mathcal{T}')$ be topological spaces with basis $\mathcal{B}$ and $\mathcal{B}'$. Then $\mathcal{T}$ is finer than $\mathcal{T}'$ if and only if for any $x\in X$, $x\in B'$, $B'\in \mathcal{B}'$, there exists $B\in \mathcal{B}$, such that $x\in B$ and $x\in B\subseteq B'$. + +### Product space + +#### Definition of cartesian product + +Let $X,Y$ be sets. The cartesian product of $X$ and $Y$ is the set of all ordered pairs $(x,y)$ where $x\in X$ and $y\in Y$, denoted by $X\times Y$. + diff --git a/content/Math4201/Math4201_L16.md b/content/Math4201/Math4201_L16.md index 2739b5f..23dfb7f 100644 --- a/content/Math4201/Math4201_L16.md +++ b/content/Math4201/Math4201_L16.md @@ -119,7 +119,7 @@ Then $Z$ is a closed subset of $(\operatorname{Map}(X,Y),\rho)$. We need to show that $\overline{Z}=Z$. -Since $\operatorname{Map}(X,Y)$ is a metric space, this is equivalent to showing that: $f_n:X\to Y\in Z$ continuous, +Since $\operatorname{Map}(X,Y)$ is a metric space, this is equivalent to showing that: Let $f_n:X\to Y\in Z$ be a sequence of continuous maps, Which is to prove the uniform convergence, @@ -127,28 +127,33 @@ $$ f_n \to f \in \operatorname{Map}(X,Y) $$ -Then we want to show that $f$ is continuous. +Then we want to show that $f$ is also continuous. -Let $B_r(y)$ be an arbitrary ball in $Y$, it suffices to show that $f^{-1}(B_r(y))$ is open in $X$. +It is to show that for any open subspace $V$ of $Y$, $f^{-1}(V)$ is open in $X$. -Take $N$ to be large enough such that for $n\geq N$, we have +Take $x_0\in f^{-1}(V)$, we'd like to show that there is an open neighborhood $U$ of $x_0$ such that $U\subseteq f^{-1}(V)$. -$$ -\rho(f_n(x), f(x)) < \frac{r}{3} -$$ +Since $x_0\in f^{-1}(V)$, then $f(x_0)\in V$. By metric definition, there is $r>0$ such that $B_r(f(x_0))\subseteq V$. -In particular, this holds for $n=N$. So we have +Take $N$ to be large enough such $\rho(f_N(x), f(x)) < \frac{r}{3}$ -$$ -d(f_N(x), f(x)) < \frac{r}{3},\forall x\in X -$$ +So $\forall x\in X$, $d(f(x),f_N(x))<\frac{r}{3}$ -Take $x_0\in f^{-1}(B_r(y))$, we'd like to show that there is an open ball around $x_0$ in $f^{-1}(B_r(y))$. +Since $f_N$ is continuous, $f_N^{-1}(B_{r/3}(f(x_0)))$ is an open set $U\subseteq X$ containing $x_0$. -Since $f_N$ is continuous, $f^{-1}_N(B_{\frac{r}{3}}(y))$ is open in $X$. +Take $x\in U$, $d(f(x),f(x_0)) \ No newline at end of file diff --git a/content/Math4201/Math4201_L18.md b/content/Math4201/Math4201_L18.md index 5b12654..b15a2d2 100644 --- a/content/Math4201/Math4201_L18.md +++ b/content/Math4201/Math4201_L18.md @@ -1,4 +1,4 @@ -# Math 4201 Topology (Lecture 18) +# Math4201 Topology I (Lecture 18) ## Quotient topology diff --git a/content/Math4201/Math4201_L2.md b/content/Math4201/Math4201_L2.md index 3048040..122619d 100644 --- a/content/Math4201/Math4201_L2.md +++ b/content/Math4201/Math4201_L2.md @@ -20,7 +20,9 @@ The elements of $\mathcal{T}$ are called **open sets**. The topological space is denoted by $(X, \mathcal{T})$. -#### Examples +
+ +Examples of topological spaces Trivial topology: Let $X$ be arbitrary. The trivial topology is $\mathcal{T}_0 = \{\emptyset, X\}$ @@ -40,12 +42,18 @@ $\mathcal{T}_2 = \{\emptyset, \{a\}, \{a,b\}\}$ $\mathcal{T}_3 = \{\emptyset, \{b\}, \{a,b\}\}$ -Non-examples: +
+ +
+ +Non-example of topological space 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 $\mathcal{T}\coloneqq \{U\subseteq X|X\setminus U \text{ is finite}\}\cup \{\emptyset\}$ diff --git a/content/Math4201/Math4201_L3.md b/content/Math4201/Math4201_L3.md index bb6e4bb..89664a3 100644 --- a/content/Math4201/Math4201_L3.md +++ b/content/Math4201/Math4201_L3.md @@ -12,7 +12,7 @@ Let $X$ be a set. A basis for a topology on $X$ is a collection $\mathcal{B}$ (e 2. $\forall B_1,B_2\in \mathcal{B}$, $\forall x\in B_1\cap B_2$, $\exists B_3\in \mathcal{B}$ such that $x\in B_3\subseteq B_1\cap B_2$
-Example 1 +Example of standard basis in real numbers Let $X=\mathbb{R}$ and $\mathcal{B}=\{(a,b)|a,b\in \mathbb{R},a
-Example 2 +Example of lower limit basis in real numbers Let $X=\mathbb{R}$ and $\mathcal{B}_{LL}=\{[a,b)|a,b\in \mathbb{R},a -Example 3 +Example of open rectangles basis for real plane Let $X=\mathbb{R}^2$ and $\mathcal{B}$ be the collection of rectangle of the form $(a,b)\times (c,d)$ where $a,b,c,d\in \mathbb{R}$ and $a
-Example 4 +Example of open disks basis for real plane Let $X=\mathbb{R}^2$ and $\mathcal{B}$ be the collection of open disks. diff --git a/content/Math4201/Math4201_L4.md b/content/Math4201/Math4201_L4.md index 5f33bc7..fdc1dac 100644 --- a/content/Math4201/Math4201_L4.md +++ b/content/Math4201/Math4201_L4.md @@ -109,7 +109,7 @@ So $B_1\cap B_2\in \mathcal{B}$.
-We call $\mathcal{B}$ the topology generated by the subbasis $\mathcal{S}$. Denote it by $\mathcal{T}_{\mathcal{S}}$. +We call the topology generated by $\mathcal{B}$ the topology generated by the subbasis $\mathcal{S}$. Denote it by $\mathcal{T}_{\mathcal{S}}$. An open set with respect to $\mathcal{T}_{\mathcal{S}}$ is a subset of $X$ such that it can be written as a union of finitely intersections of elements of $\mathcal{S}$. diff --git a/content/Math4201/Math4201_L5.md b/content/Math4201/Math4201_L5.md index 2f4eab6..215f917 100644 --- a/content/Math4201/Math4201_L5.md +++ b/content/Math4201/Math4201_L5.md @@ -51,7 +51,7 @@ So the discrete topology is finer than the trivial topology. Let $\mathcal{T}$ and $\mathcal{T}'$ be topologies on $X$ associated with bases $\mathcal{B}$ and $\mathcal{B}'$. Then $$ -\mathcal{T}\text{ is finer than } \mathcal{T}'\iff \text{ for any } B'\in \mathcal{B}', \exists B\in \mathcal{B} \text{ such that } B'\subseteq B +\mathcal{T}\text{ is finer than } \mathcal{T}'\iff \text{ for any } x\in X, x\in B'\in \mathcal{B}', \exists B\in \mathcal{B} \text{ such that } x\in B\subseteq B' $$