From 74dcdc04dc8774a5a9c0c796c4cd7bedb94a143d Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 9 Oct 2025 23:53:23 -0500 Subject: [PATCH] partial updates for review, and fix typos --- content/Math4201/Exam_reviews/Math4201_E1.md | 215 +++++++++++++++++++ content/Math4201/Math4201_L14.md | 2 +- content/Math4201/Math4201_L15.md | 8 +- content/Math4201/Math4201_L17.md | 3 +- content/Math4201/Math4201_L9.md | 2 +- 5 files changed, 224 insertions(+), 6 deletions(-) diff --git a/content/Math4201/Exam_reviews/Math4201_E1.md b/content/Math4201/Exam_reviews/Math4201_E1.md index dcfedc6..9679b63 100644 --- a/content/Math4201/Exam_reviews/Math4201_E1.md +++ b/content/Math4201/Exam_reviews/Math4201_E1.md @@ -32,6 +32,12 @@ $U\subseteq X$ is an open set if $U\in \mathcal{T}$ $Z\subseteq X$ is a closed set if $X\setminus Z\in \mathcal{T}$ +> [!WARNING] +> +> A set is closed is not the same as its not open. +> +> In all topologies over non-empty sets, $X, \emptyset$ are both closed and open. + ### Basis #### Definition of topological basis @@ -91,3 +97,212 @@ Let $(X,\mathcal{T})$ and $(X,\mathcal{T}')$ be topological spaces with basis $\ 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 $X\times Y$. +#### Definition of product topology + +Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. Then the product topology on $X\times Y$ is the topology generated by the basis + +$$ +\mathcal{B}_{X\times Y}=\{U\times V, U\in \mathcal{T}_X, V\in \mathcal{T}_Y\} +$$ + +or equivalently, + +$$ +\mathcal{B}_{X\times Y}'=\{U\times V, U\in \mathcal{B}_X, V\in \mathcal{B}_Y\} +$$ + +> Product topology generated from open sets of $X$ and $Y$ is the same as product topology generated from their corresponding basis + +### Subspace topology + +#### Definition of subspace topology + +Let $(X,\mathcal{T})$ be a topological space and $Y\subseteq X$. Then the subspace topology on $Y$ is the topology given by + +$$ +\mathcal{T}_Y=\{U\cap Y|U\in \mathcal{T}\} +$$ + +or equivalently, let $\mathcal{B}$ be the basis for $(X,\mathcal{T})$. Then the subspace topology on $Y$ is the topology generated by the basis + +$$ +\mathcal{B}_Y=\{U\cap Y| U\in \mathcal{B}\} +$$ + +#### Lemma of open sets in subspace topology + +Let $(X,\mathcal{T})$ be a topological space and $Y\subseteq X$. Then if $U\subseteq Y$, $U$ is open in $(Y,\mathcal{T}_Y)$, then $U$ is open in $(X,\mathcal{T})$. + +> This also holds for closed set in closed subspace topology + +### Interior and closure + +#### Definition of interior + +The interior of $A$ is the largest open subset of $A$. + +$$ +A^\circ=\bigcup_{U\subseteq A, U\text{ is open in }X} U +$$ + +#### Definition of closure + +The closure of $A$ is the smallest closed superset of $A$. + +$$ +\overline{A}=\bigcap_{U\supseteq A, U\text{ is closed in }X} U +$$ + +#### Definition of neighborhood + +A neighborhood of a point $x\in X$ is an open set $U\in \mathcal{T}$ such that $x\in U$. + +#### Definition of limit points + +A point $x\in X$ is a limit point of $A$ if every neighborhood of $x$ contains a point in $A-\{x\}$. + +We denote the set of all limits points of $A$ by $A'$. + +$\overline{A}=A\cup A'$ + +### Sequences and continuous functions + +#### Definition of convergence + +Let $X$ be a topological space. A sequence $(x_n)_{n\in\mathbb{N}_+}$ in $X$ converges to $x\in X$ if for any neighborhood $U$ of $x$, there exists $N\in\mathbb{N}_+$ such that $\forall n\geq N, x_n\in U$. + +#### Definition of Hausdoorff space + +A topological space $(X,\mathcal{T})$ is Hausdorff if for any two distinct points $x,y\in X$, there exist open neighborhoods $U$ and $V$ of $x$ and $y$ respectively such that $U\cap V=\emptyset$. + +#### Uniqueness of convergence in Hausdorff spaces + +In a Hausdorff space, if a sequence $(x_n)_{n\in\mathbb{N}_+}$ converges to $x\in X$ and $y\in X$, then $x=y$. + +#### Closed singleton in Hausdorff spaces + +In a Hausdorff space, if $x\in X$, then $\{x\}$ is a closed set. + +#### Definition of continuous function + +Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. A function $f:X\to Y$ is continuous if for any open set $U\subseteq Y$, $f^{-1}(U)$ is open in $X$. + +#### Definition of point-wise continuity + +Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. A function $f:X\to Y$ is point-wise continuous at $x\in X$ if for every openset $V\subseteq Y$, $f(x)\in V$ then there exists an open set $U\subseteq X$ such that $x\in U$ and $f(U)\subseteq V$. + +#### Lemma of continuous functions + +If $f:X\to Y$ is point-wise continuous for all $x\in X$, then $f$ is continuous. + +#### Properties of continuous functions + +If $f:X\to Y$ is continuous, then + +1. $\forall A\subseteq Y$, $f^{-1}(A^c)=X\setminus f^{-1}(A)$ (complements maps to complements) +2. $\forall A_\alpha\subseteq Y, \alpha\in I$, $f^{-1}(\bigcup_{\alpha\in I} A_\alpha)=\bigcup_{\alpha\in I} f^{-1}(A_\alpha)$ +3. $\forall A_\alpha\subseteq Y, \alpha\in I$, $f^{-1}(\bigcap_{\alpha\in I} A_\alpha)=\bigcap_{\alpha\in I} f^{-1}(A_\alpha)$ +4. $f^{-1}(U)$ is open in $X$ for any open set $U\subseteq Y$. +5. $f$ is continuous at $x\in X$. +6. $f^{-1}(V)$ is closed in $X$ for any closed set $V\subseteq Y$. +7. Assume $\mathcal{B}$ is a basis for $Y$, then $f^{-1}(\mathcal{B})$ is open in $X$ for any $B\in \mathcal{B}$. +8. $\forall A\subseteq X$, $\overline{f(A)}=f(\overline{A})$ + +#### Definition of homeomorphism + +Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces. A function $f:X\to Y$ is a homeomorphism if $f$ is continuous, bijective and $f^{-1}:Y\to X$ is continuous. + +#### Ways to construct continuous functions + +1. If $f:X\to Y$ is constant function, $f(x)=y_0$ for all $x\in X$, then $f$ is continuous. (constant functions are continuous) +2. If $A$ is a subspace of $X$, $f:A\to X$ is the inclusion map $f(x)=x$ for all $x\in A$, then $f$ is continuous. (inclusion maps are continuous) +3. If $f:X\to Y$ is continuous, $g:Y\to Z$ is continuous, then $g\circ f:X\to Z$ is continuous. (composition of continuous functions is continuous) +4. If $f:X\to Y$ is continuous, $A$ is a subspace of $X$, then $f|_A:X\to Y$ is continuous. (domain restriction is continuous) +5. If $f:X\to Y$ is continuous, $Z$ is a subspace of $Y$, then $f:X\to Z$, $g(x)=f(x)\cap Z$ is continuous. If $Y$ is a subspace of $Z$, then $h:X\to Z$, $h(x)=f(x)$ is continuous (composition of $f$ and inclusion map). +6. If $f:X\to Y$ is continuous, $X$ can be written as a union of open sets $\{U_\alpha\}_{\alpha\in I}$, then $f|_{U_\alpha}:X\to Y$ is continuous. +7. If $X=Z_1\cup Z_2$, and $Z_1,Z_2$ are closed equipped with subspace topology, let $g_1:Z_1\to Y$ and $g_2:Z_2\to Y$ be continuous, and for all $x\in Z_1\cap Z_2$, $g_1(x)=g_2(x)$, then $f:X\to Y$ by $f(x)\begin{cases}g_1(x), & x\in Z_1 \\ g_2(x), & x\in Z_2\end{cases}$ is continuous. (pasting lemma) +8. $f:X\to Y$ is continuous, $g:X\to Z$ is continuous if and only if $H:X\to Y\times Z$, where $Y\times Z$ is equipped with the product topology, $H(x)=(f(x),g(x))$ is continuous. (proved in homework) + +### Metric spaces + +#### Definition of metric + +A metric on $X$ is a function $d:X\times X\to \mathbb{R}$ such that $\forall x,y\in X$, + +1. $d(x,x)=0$ +2. $d(x,y)\geq 0$ +3. $d(x,y)=d(y,x)$ +4. $d(x,y)+d(y,z)\geq d(x,z)$ + +#### Definition of metric ball + +The metric ball $B_r^{d}(x)$ is the set of all points $y\in X$ such that $d(x,y)\leq r$. + +#### Definition of metric topology + +Let $X$ be a metric space with metric $d$. Then $X$ is equipped with the metric topology generated by the metric balls $B_r^{d}(x)$ for $r>0$. + +#### Definition of metrizable + +A topological space $(X,\mathcal{T})$ is metrizable if it is the metric topology for some metric $d$ on $X$. + +#### Hausdorff axiom for metric spaces + +Every metric space is Hausdorff (take metric balls $B_r(x)$ and $B_r(y)$, $r=\frac{d(x,y)}{2}$). + +If a topology isn't Hausdorff, then it isn't metrizable. + +Prove by triangle inequality and contradiction. + +#### Common metrics in $\mathbb{R}^n$ + +Euclidean metric + +$$ +d(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2} +$$ + +Square metric + +$$ +\rho(x,y)=\max_{i=1}^n |x_i-y_i| +$$ + +Manhattan metric + +$$ +m(x,y)=\sum_{i=1}^n |x_i-y_i| +$$ + +These metrics are equivalent. + +#### Product topology and metric + +If $(X,d),(Y,d')$ are metric spaces, then $X\times Y$ is metric space with metric $d(x,y)=\max\{d(x_1,y_1),d(x_2,y_2)\}$. + +#### Uniform metric + +Let $\mathbb{R}^\omega$ be the set of all infinite sequences of real numbers. Then $\overline{d(x,y)}=\sup_{i=1}^\omega \min\{1,|x_i-y_i|\}$, the uniform metric on $\mathbb{R}^\omega$ is a metric. + +#### Metric space and converging sequences + +Let $X$ be a topological space, $A\subseteq X$, $x_n\to x$ such that $x_n\in A$. Then $x\in \overline{A}$. + +If $X$ is a **metric space**, $A\subseteq X$, $x\in \overline{A}$, then there exists converging sequence $x_n\to x$ such that $x_n\in A$. + +#### First countability axiom + +A topological space $(X,\mathcal{T})$ satisfies the first countability axiom if any point $x\in X$, there is a sequence of open neighborhoods of $x$, $\{V_n\}_{n=1}^\infty$ such that any open neighborhood $U$ of $x$ contains one of $V_n$. + +Apply the theorem above, we have if $(X,\mathcal{T})$ satisfies the first countability axiom, then every convergent sequence converges to a point in the closure of the sequence. + +### Metric defined for functions + +#### Definition for bounded metric space + +A metric space $(Y,d)$ is bounded if there is $M\in \mathbb{R}^{\geq 0}$ such that for all $y_1,y_2\in Y$, $d(y_1,y_2)\leq M$. + +#### Definition for metric defined for functions + +Let $X$ be a topological space and $Y$ be a bounded metric space, then the set of all maps, denoted by $\operatorname{Map}(X,Y)$, $f:X\to Y\in \operatorname{Map}(X,Y)$ is a metric space with metric $\rho(f,g)=\sup_{x\in X} d(f(x),g(x))$. + diff --git a/content/Math4201/Math4201_L14.md b/content/Math4201/Math4201_L14.md index 19c1e35..69d42a7 100644 --- a/content/Math4201/Math4201_L14.md +++ b/content/Math4201/Math4201_L14.md @@ -2,7 +2,7 @@ ## Metric topology -### Subspace topology and metric topology +### Product topology and metric topology If $X$ and $Y$ are metrizable spaces, then the product space $X\times Y$ is metrizable. diff --git a/content/Math4201/Math4201_L15.md b/content/Math4201/Math4201_L15.md index 5b490da..197bc01 100644 --- a/content/Math4201/Math4201_L15.md +++ b/content/Math4201/Math4201_L15.md @@ -72,9 +72,13 @@ Find an example of a function $f:X\to Y$ which is not continuous but for any con
Solution -Let $f:S^1\to [0,1)$ be defined by $f(x,y)=\sin^{-1}(\frac{y}{x})$. +Consider $X=\mathbb{R}$ with complement finite topology and $Y=\mathbb{R}$ with the standard topology. -This is not continuous because $[0,1)$ +Take identity function $f(x)=x$. + +This function is not continuous by trivially taking $(0,1)\subseteq \mathbb{R}$ and the complement of $(0,1)$ is not a finite set, so the function is not continuous. + +However, for every convergent sequence in $X$, $\{x_n\}_{n=1}^\infty\to x$, the sequence $\{f(x_n)\}_{n=1}^\infty\to f(x)$ trivially.
diff --git a/content/Math4201/Math4201_L17.md b/content/Math4201/Math4201_L17.md index fa181d8..183a27a 100644 --- a/content/Math4201/Math4201_L17.md +++ b/content/Math4201/Math4201_L17.md @@ -83,5 +83,4 @@ $$ $$ q^{-1}(\bigcap_{\alpha \in I} U_\alpha)=\bigcap_{\alpha \in I} q^{-1}(U_\alpha) $$ - - \ No newline at end of file + diff --git a/content/Math4201/Math4201_L9.md b/content/Math4201/Math4201_L9.md index 9ff68a4..83df526 100644 --- a/content/Math4201/Math4201_L9.md +++ b/content/Math4201/Math4201_L9.md @@ -2,7 +2,7 @@ ## Convergence of sequences -Let $X$ be a topological space and $\{x_n\}_{n\in\mathbb{N}_+}$ be a sequence of points in $X$. WE say $x_n\to x$ as $n\to \infty$ ($x_n$ converges to $x$ as $n\to \infty$) +Let $(X,\mathcal{T})$ be a topological space and $\{x_n\}_{n\in\mathbb{N}_+}$ be a sequence of points in $X$. We say $x_n\to x$ as $n\to \infty$ ($x_n$ converges to $x$ as $n\to \infty$) if for any open neighborhood $U$ of $x$, there exists $N\in\mathbb{N}_+$ such that $\forall n\geq N, x_n\in U$.