拓扑第二课-ZhiMap思维导图

拓扑第二课
进入思维导图模式
- The subspace topology
  （子空间拓扑）
  - Def
    - If Y is a subset of X, then $T_Y=\{\ Y\cap U\mid U\in T_X\}$ is a topology on Y, called the subspce topology. $(Y,T_Y)$ is called a subspace of $(X,T_X)$ .
  - Lemma
    - If $\mathscr B$ is a basis for $(X,T_X)$ , then $\mathscr B_Y=\{B \cap Y \mid B \in \mathscr B\}$ is a basis for the subspace topology on Y.
    - $U \in T_Y , Y \subseteq X \nRightarrow U \in T_X$
    - $U \in T_Y , Y \in T_X \Rightarrow U \in T_X$ （“开”的传递性）
  - Theorem
    - If A is a subspace of X, B is a subspace of Y, then the product topology on A×B inherits as a subspace of X×Y.
- Topology and Basis
  （拓扑和基）
  - Topology T
    - Def（对“开”这一性质的规定）
      - (1) $\emptyset,X \in T$ （空集、全集必开）
      - (2) The union of any collection of elements of T is in T （开集之并必开）
      - (3) The intersection of any finite collection of elements of T is in T
        （开集之有限交必开）
    - (X,T) is called a topological space
    - $u \in T$ ,by convention,call u an open
  - Basis $\mathscr B$
    - Def
      - (1) $\forall x \in X, \exists B \in \mathscr B\ and \ \ x\in B$ （B们可以覆盖X）
      - (2) $\forall B_1,B_2 \in \mathscr B,\ \text{if \ }x \in B_1 \cap B_2, \text{ then }\ \exists B_3 \in \mathscr B,\ s.t. \ x \in B_3 \subset B_1\cap B_2$ （B们可以精确覆盖某两个B的交集）
    - $\mathscr B$ defines a topology $T_\mathscr B$ on X
      - $\forall U \subseteq X,\ \ U\in T_\mathscr B \ \text{iff }\ \forall x\in U, \exists B \in \mathscr B\ s.t.\ x\in B\subseteq U$
        （B们可以精准覆盖U $\Leftrightarrow$ U是开集）
  - Properties
    - A basis is not unique for a given topology $T_\mathscr B$
      （某个“开”的定义并非只能由一个基生成）
    - $\mathscr B \subseteq T_\mathscr B$ （B们都是开集）（因为B自己就能精准覆盖B自己）
    - $T_\mathscr B$ equals the collections of all unions of elements of $\mathscr B$
      （“开”由 $\mathscr B$ 定义 $\Leftrightarrow$ 开集之集就是任意些B的并集之集）
    - Suppose $\mathscr B$ is a collection of open sets of X,then $\mathscr B$ is a basis for T if
      - $\forall U\in T\text{ and } x\in U,\exists B \in \mathscr B\text{ s.t. } x\in B \subseteq U$
        (所有开集都可以被B们精准覆盖）
  - Subbasis（子基） S $\mathscr S$
    - Def
      - a collection of subsets of X, whose union is X（X的某个覆盖）
    - Theorem
      - topology $T_S$ generated by S is defined to be the collection of all unions of finite intersections of elements in S
        （ $T_S$ 由S的全部有限交集之各种并集组成）
    - Remark
      - Collection of all finite intersection of elements in S is a basis
- Closed sets
  (闭集）
  - Def
    - If $A \subseteq X,X\setminus A \in T_X$ call A a closed set
  - Theorem
    - (1) ∅，X are closed （空集、全集必闭）
    - (2) Arbitrary intersection of closed sets is closed（闭集之交必闭）
    - (3) Finite unions of closed sets are closed（闭集之有限并必闭）
    - A is closed in $Y(\subseteq X)$ $Y(\subseteq X) \Leftrightarrow A=C\cap Y$ ,C is closed in X
      (Y中闭集之集=X中闭集交Y之集）
    - If A is closed in Y, and Y is closed in X, then A is closed in X.
      （“闭”的传递性）
- The product topology on $X \times Y$
  (乘积拓扑）
  - Def
    - the topology having $\mathscr B=\{ \ U \times V\ |\ U \in T_X,V \in T_Y\}$ as basis
  - Theorem
    - If $\mathscr B_1$ and $\mathscr B_2$ are basis of $(X_1,T_1),(X_2,T_2)$ respectively, then $\mathscr B'=\{B_1 \times B_2\ |\ B_1 \in \mathscr B_1,B_2 \in \mathscr B_2\}$ is a basis for the product topology.（B们可精准覆盖的=B'们可精准覆盖的）
  - Projection maps
    （投影函数）
    - Def
      - $\Pi_1:(x,y)\mapsto x,\ \Pi_2:(x,y)\mapsto y$
        $\Pi_1^{-1}(U)=\{(x,y)\ |\ \Pi_1(x,y) \in U\}=U\times Y$
      - $\Pi_2:(x,y)\mapsto y$
        $\Pi_2^{-1}(V)=\{(x,y)\ |\ \Pi_2(x,y) \in V\}=X\times V$
    - Theorem
      - $S=\{\ \Pi_1^{-1}(U)\mid U \in T_X\}\cup\{\ \Pi_2^{-1}(V)\mid V \in T_Y\}$
        is a subbasis for the products topology
      - $\mathcal S=$ $\{\ U \times Y \mid U \in \mathscr B_X\}\cup\{\ X \times V \mid V \in \mathscr B_Y\}$ 也可以
      - $\mathcal S=\{U\times Y\mid U\in\mathcal S_X\} \cup \{X\times V\mid V\in\mathcal S_Y\}$ 也可以
- Closure and interior of a set
  （闭包与内点集）
  - Def（ $A \subseteq X$ ）
    - The interior of A( $\text{Int }A$ )
      - the union of all open sets in X contianed in A.
        （包含于A的开集之并）（A中能被开集覆盖的部分）
    - The closure of A( $\overline A$ )
      - the intersection of all closed sets in X contianing A.
        （包含A的闭集之交）
      - Theorem
        1) $x \in \overline A$ iff $\forall x \in U \in T, \, U \cap A \neq \emptyset$
        2) $x \in \overline A$ iff $\forall x \in B \in \mathscr B, \, B \cap A \neq \emptyset$
  - Properties
    - $\text{Int }A \subseteq A \subseteq \overline A$
    - If A is open, A= $\text{Int }A$
    - If A is closed, A= $\overline A$
  - Theorem
    - $A \subseteq Y\subseteq X \Rightarrow \overline A_Y =\overline A_X \cap Y$
      （A在Y中的聚点必是A在X中的聚点）

拓扑第二课

​The subspace topology （子空间拓扑）

​Def

​If Y is a subset of X, then TY={ Y∩U∣U∈TX}T_Y=\{\ Y\cap U\mid U\in T_X\}TY​={ Y∩U∣U∈TX​} is a topology on Y, called the subspce topology. (Y,TY)(Y,T_Y)(Y,TY​) is called a subspace of (X,TX)(X,T_X)(X,TX​).

​Lemma

​If B\mathscr BB is a basis for (X,TX)(X,T_X)(X,TX​) , then BY={B∩Y∣B∈B}\mathscr B_Y=\{B \cap Y \mid B \in \mathscr B\}BY​={B∩Y∣B∈B} is a basis for the subspace topology on Y.

​ U∈TY,Y⊆X⇏U∈TXU \in T_Y , Y \subseteq X \nRightarrow U \in T_XU∈TY​,Y⊆X⇏U∈TX​

U∈TY,Y∈TX⇒U∈TXU \in T_Y , Y \in T_X \Rightarrow U \in T_XU∈TY​,Y∈TX​⇒U∈TX​ （“开”的传递性）

​Theorem

​If A is a subspace of X, B is a subspace of Y, then the product topology on A×B inherits as a subspace of X×Y.

​Topology and Basis （拓扑和基）

​Topology T

​Def（对“开”这一性质的规定）

​(1) ∅,X∈T\emptyset,X \in T∅,X∈T （空集、全集必开）

​(2) The union of any collection of elements of T is in T （开集之并必开）

​(3) The intersection of any finite collection of elements of T is in T（开集之有限交必开）

​(X,T) is called a topological space

​ u∈Tu \in Tu∈T ,by convention,call u an open

​Basis B\mathscr BB

​Def

​(1) ∀x∈X,∃B∈B and x∈B\forall x \in X, \exists B \in \mathscr B\ and \ \ x\in B∀x∈X,∃B∈B and x∈B （B们可以覆盖X）

B\mathscr BB defines a topology TBT_\mathscr BTB​ on X

Properties

​A basis is not unique for a given topology TBT_\mathscr BTB​ （某个“开”的定义并非只能由一个基生成）

​ B⊆TB\mathscr B \subseteq T_\mathscr BB⊆TB​ （B们都是开集）（因为B自己就能精准覆盖B自己）

​ TBT_\mathscr BTB​ equals the collections of all unions of elements of B\mathscr BB （“开”由 B\mathscr BB 定义 ⇔\Leftrightarrow⇔ 开集之集就是任意些B的并集之集）

​Suppose B\mathscr BB is a collection of open sets of X,then B\mathscr BB is a basis for T if

∀U∈T and x∈U,∃B∈B s.t. x∈B⊆U\forall U\in T\text{ and } x\in U,\exists B \in \mathscr B\text{ s.t. } x\in B \subseteq U∀U∈T and x∈U,∃B∈B s.t. x∈B⊆U (所有开集都可以被B们精准覆盖）

​Subbasis（子基） S S\mathscr S

​Def

​a collection of subsets of X, whose union is X（X的某个覆盖）

​Theorem

​topology TST_STS​ generated by S is defined to be the collection of all unions of finite intersections of elements in S（ TST_STS​ 由S的全部有限交集之各种并集组成）

​Remark

Collection of all finite intersection of elements in S is a basis

​Closed sets (闭集）

​Def

​If A⊆X,X∖A∈TXA \subseteq X,X\setminus A \in T_XA⊆X,X∖A∈ T ,call A a closed set

​Theorem

​(1) ∅，X are closed （空集、全集必闭）

​(2) Arbitrary intersection of closed sets is closed（闭集之交必闭）

​(3) Finite unions of closed sets are closed（闭集之有限并必闭）

A is closed in Y(⊆X)Y(\subseteq X) Y(⊆X)⇔A=C∩YY(\subseteq X) \Leftrightarrow A=C\cap YY(⊆X)⇔A=C∩Y ,C is closed in X(Y中闭集之集=X中闭集交Y之集）

​If A is closed in Y, and Y is closed in X, then A is closed in X.（“闭”的传递性）

​The product topology on X×YX \times YX×Y (乘积拓扑）

​Def

​the topology having ​ B={ U×V ∣ U∈TX,V∈TY}\mathscr B=\{ \ U \times V\ |\ U \in T_X,V \in T_Y\}B={ U×V ∣ U∈TX​,V∈TY​} as basis

​Theorem

​Projection maps （投影函数）

​Def

​ Π1:(x,y)↦x, Π2:(x,y)↦y\Pi_1:(x,y)\mapsto x,\ \Pi_2:(x,y)\mapsto yΠ1​:(x,y)↦x

​ Π1−1(U)={(x,y) ∣ Π1(x,y)∈U}=U×Y\Pi_1^{-1}(U)=\{(x,y)\ |\ \Pi_1(x,y) \in U\}=U\times YΠ1−1​(U)={(x,y) ∣ Π1​(x,y)∈U}=U×Y

​ Π2:(x,y)↦y\Pi_2:(x,y)\mapsto yΠ2​:(x,y)↦y

​ Π2−1(V)={(x,y) ∣ Π2(x,y)∈V}=X×V\Pi_2^{-1}(V)=\{(x,y)\ |\ \Pi_2(x,y) \in V\}=X\times VΠ2−1​(V)={(x,y) ∣ Π2​(x,y)∈V}=X×V

​Theorem

S=\{\ \Pi_1^{-1}(U)\mid U \in T_X\}\cup\{\ \Pi_2^{-1}(V)\mid V \in T_Y\}S={ Π1−1​(U)∣U∈TX​}∪{ Π2−1​(V)∣V∈TY​} is a subbasis for the products topology

​ S=\mathcal S=S= { U×Y∣U∈BX}∪{ X×V∣V∈BY}\{\ U \times Y \mid U \in \mathscr B_X\}\cup\{\ X \times V \mid V \in \mathscr B_Y\}{ U×Y∣U∈BX​}∪{ X×V∣V∈BY​} 也可以

​ S={U×Y∣U∈SX}∪{X×V∣V∈SY}\mathcal S=\{U\times Y\mid U\in\mathcal S_X\} \cup \{X\times V\mid V\in\mathcal S_Y\}S={U×Y∣U∈SX​}∪{X×V∣V∈SY​} 也可以

​Closure and interior of a set （闭包与内点集）

​Def（ A⊆XA \subseteq XA⊆X ）

​The interior of A( Int A \text{Int }AInt A )

the union of all open sets in X contianed in A. （包含于A的开集之并）（A中能被开集覆盖的部分）

​The closure of A( A‾\overline AA )

​the intersection of all closed sets in X contianing A.（包含A的闭集之交）

​Theorem

1) x∈A‾x \in \overline Ax∈A iff​ ∀x∈U∈T, U∩A̸=∅\forall x \in U \in T, \, U \cap A \neq \emptyset∀x∈U∈T,U∩A̸=∅

​2) x∈A‾x \in \overline Ax∈A iff ∀x∈B∈B, B∩A̸=∅\forall x \in B \in \mathscr B, \, B \cap A \neq \emptyset∀x∈B∈B,B∩A̸=∅

Properties

​ Int A⊆A⊆A‾ \text{Int }A \subseteq A \subseteq \overline AInt A⊆A⊆A

​If A is open, A= Int A \text{Int }AInt A

​If A is closed, A= A‾\overline AA

​Theorem

​ A⊆Y⊆X⇒A‾Y=A‾X∩YA \subseteq Y\subseteq X \Rightarrow \overline A_Y =\overline A_X \cap YA⊆Y⊆X⇒AY​=AX​∩Y （A在Y中的聚点必是A在X中的聚点）

The subspace topology
（子空间拓扑）

Def

If Y is a subset of X, then $T_Y=\{\ Y\cap U\mid U\in T_X\}$ is a topology on Y, called the subspce topology. $(Y,T_Y)$ is called a subspace of $(X,T_X)$ .

Lemma

If $\mathscr B$ is a basis for $(X,T_X)$ , then $\mathscr B_Y=\{B \cap Y \mid B \in \mathscr B\}$ is a basis for the subspace topology on Y.

$U \in T_Y , Y \subseteq X \nRightarrow U \in T_X$

$U \in T_Y , Y \in T_X \Rightarrow U \in T_X$ （“开”的传递性）

Theorem

If A is a subspace of X, B is a subspace of Y, then the product topology on A×B inherits as a subspace of X×Y.

Topology and Basis
（拓扑和基）

Topology T

Def（对“开”这一性质的规定）

(1) $\emptyset,X \in T$ （空集、全集必开）

(2) The union of any collection of elements of T is in T （开集之并必开）

(3) The intersection of any finite collection of elements of T is in T
（开集之有限交必开）

(X,T) is called a topological space

$u \in T$ ,by convention,call u an open

Basis $\mathscr B$

Def

(1) $\forall x \in X, \exists B \in \mathscr B\ and \ \ x\in B$ （B们可以覆盖X）

$\mathscr B$ defines a topology $T_\mathscr B$ on X

A basis is not unique for a given topology $T_\mathscr B$
（某个“开”的定义并非只能由一个基生成）

$\mathscr B \subseteq T_\mathscr B$ （B们都是开集）（因为B自己就能精准覆盖B自己）

$T_\mathscr B$ equals the collections of all unions of elements of $\mathscr B$
（“开”由 $\mathscr B$ 定义 $\Leftrightarrow$ 开集之集就是任意些B的并集之集）

Suppose $\mathscr B$ is a collection of open sets of X,then $\mathscr B$ is a basis for T if

$\forall U\in T\text{ and } x\in U,\exists B \in \mathscr B\text{ s.t. } x\in B \subseteq U$
(所有开集都可以被B们精准覆盖）

Subbasis（子基） S $\mathscr S$

Def

a collection of subsets of X, whose union is X（X的某个覆盖）

Theorem

topology $T_S$ generated by S is defined to be the collection of all unions of finite intersections of elements in S
（ $T_S$ 由S的全部有限交集之各种并集组成）

Remark

Closed sets
(闭集）

Def

If $A \subseteq X,X\setminus A \in T_X$ call A a closed set

Theorem

(1) ∅，X are closed （空集、全集必闭）

(2) Arbitrary intersection of closed sets is closed（闭集之交必闭）

(3) Finite unions of closed sets are closed（闭集之有限并必闭）

A is closed in $Y(\subseteq X)$ $Y(\subseteq X) \Leftrightarrow A=C\cap Y$ ,C is closed in X
(Y中闭集之集=X中闭集交Y之集）

If A is closed in Y, and Y is closed in X, then A is closed in X.
（“闭”的传递性）

The product topology on $X \times Y$
(乘积拓扑）

Def

the topology having $\mathscr B=\{ \ U \times V\ |\ U \in T_X,V \in T_Y\}$ as basis

Theorem

Projection maps
（投影函数）

Def

$\Pi_1:(x,y)\mapsto x,\ \Pi_2:(x,y)\mapsto y$

$\Pi_1^{-1}(U)=\{(x,y)\ |\ \Pi_1(x,y) \in U\}=U\times Y$

$\Pi_2:(x,y)\mapsto y$

$\Pi_2^{-1}(V)=\{(x,y)\ |\ \Pi_2(x,y) \in V\}=X\times V$

Theorem

$S=\{\ \Pi_1^{-1}(U)\mid U \in T_X\}\cup\{\ \Pi_2^{-1}(V)\mid V \in T_Y\}$
is a subbasis for the products topology

$\mathcal S=$ $\{\ U \times Y \mid U \in \mathscr B_X\}\cup\{\ X \times V \mid V \in \mathscr B_Y\}$ 也可以

$\mathcal S=\{U\times Y\mid U\in\mathcal S_X\} \cup \{X\times V\mid V\in\mathcal S_Y\}$ 也可以

Closure and interior of a set
（闭包与内点集）

Def（ $A \subseteq X$ ）

The interior of A( $\text{Int }A$ )

the union of all open sets in X contianed in A.
（包含于A的开集之并）（A中能被开集覆盖的部分）

The closure of A( $\overline A$ )

the intersection of all closed sets in X contianing A.
（包含A的闭集之交）

Theorem

1) $x \in \overline A$ iff $\forall x \in U \in T, \, U \cap A \neq \emptyset$

2) $x \in \overline A$ iff $\forall x \in B \in \mathscr B, \, B \cap A \neq \emptyset$

$\text{Int }A \subseteq A \subseteq \overline A$

If A is open, A= $\text{Int }A$

If A is closed, A= $\overline A$

Theorem

$A \subseteq Y\subseteq X \Rightarrow \overline A_Y =\overline A_X \cap Y$
（A在Y中的聚点必是A在X中的聚点）