partial updates for review, and fix typos

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 $\rho(f,g)=\sup_{x\in X} d(f(x),g(x))$.
+

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.
 

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
 <details>
 <summary>Solution</summary>
 
-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.
 
 </details>
 

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)
 $$
-
+</details>
-</details>

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 $\forall n\geq N, x_n\in U$.