拓扑第四课

• ​Product space revisit

  • ​Let  ${ \displaystyle \prod_{\alpha \in J} X_\alpha }$  represent the cartesian product (笛卡尔积) of  $\{ X_\alpha \}_{\alpha \in J}$ 

  • ​1.Box topology

    • basis​ $\mathscr B =\{ \,\prod_{\alpha \in J}U_\alpha \mid U_\alpha \in T_{X_\alpha}\}$ 

  • ​2.Product topology

    • ​subbasis $\mathcal S=\{ \Pi^{-1}_\beta(U_\beta) \mid \beta \in J, U_\beta \in T_{X_\beta} \}$ 

  • 当 $\{X_\alpha\}_{\alpha \in J}$ 为有限集时，1与2一致；一般情况下，1细于2。

  • ​Theorem

    • ​Both 1 and 2, if  $A_a$ is a subspace of  $X_a$ , $\forall a\in J, \prod_{a \in J}A_a$ is a subspace of  $\prod_{a \in J} X_J$ .

    • Both 1 and 2, if each space  $X_a$ is Hausdorff, $\prod_{a \in J} X_a$ is Hausdorff.

      • ​why 2

    • ​Both 1 and 2, if  $A_a \subseteq X_a, \forall a \in J$ ,then  $\prod_{a \in J} \overline{A_a}=\overline{\prod_{a \in J} A_a}$ 

      • ​why

    • ​Only 2. Let  $\displaystyle f:A\rightarrow\prod_{a\in J} X_a$ be given by  $\displaystyle a \mapsto (f_\alpha(a))_{\alpha \in J}$ ,while  $\displaystyle f_\alpha:A\rightarrow X_\alpha$ is called the coordinate function.Then  $f$ is continuous iff.  $f_\alpha$ is continuous, $\forall \alpha \in J$ .

      • ​ $\displaystyle f^{-1}(\pi_\beta^{-1}(U))=(\pi_\beta \circ f)^{-1}(U)=f_\beta^{-1}(U)$ 

      • ​When 1, try to find contradiction by  $f(x)=(x,x,...,x)$

• Quotient topology
         (商拓扑)

  • ​quotient map

    • ​def

      • ​A surjective map  $\displaystyle p:X\rightarrow Y$ is said to be a quotient map if
        a subset  $\displaystyle U \subseteq Y$ is open in Y iff.  $\displaystyle p^{-1}(U)$ is open in X.

    • ​a quotient map is sure to be continuous

  • ​open map

    • ​def

      • ​each open set maps to an open set

  • ​close map

    • ​def

      • ​each closed set maps to an closed set

  • ​Theorem

    • If a surjective cont. map p is open or closed, p is a quotient map.

    • ​Bijective quotient map is a homeomorphism

    • ​A is a subset of X, then a retraction of X onto A is a cont. map s.t. r(a)=a, it's a quotient map.

    • ​composition（复合） of quotient maps is quotient

  • ​Def

    • ​If  $\displaystyle p:X \rightarrow A$ is a surjection, then there is exactly one topology to make p a quotient map.It's called the quotient topology induced by p.

    • ​Just make sure that $\displaystyle \forall U \subseteq A, U\in T_A \Leftrightarrow p^{-1}(U)\in T_X$ 

  • ​quotient space

    • Let​ $\displaystyle Xl^*$ be a partition(分划) of X,p is the cannonical projection(正则映射)

    • The quotient topology induced by p on $\displaystyle X^*$ is called a quotient space