拓扑第九课

• ​Theorem

  • ​ $\overline{d}(a,b)=\min \{ |a-b|,1 \}$ is called the standard bounded metric on R

  • ​ $\forall x,y \in \mathbb R^\omega$ ,we can define  $\displaystyle D(x,y)=\sup_{i\in \mathbb N_+}\{\frac{\overline d(x_i,y_i)}{i}\}$ as a metric on which gives the product topology on  $\displaystyle \mathbb R^\omega$ 

• ​Inbedding theorem

  • ​Let X be a space with T(1) condition

  • ​suppose  $\displaystyle \{f_\alpha\}_{\alpha \in J}$ is a family of continuous maps  $\displaystyle f_\alpha:X \rightarrow \mathbb R$ satisfying that for all x0 in X and each open neighborhood U of x0, there is an index n so that fn is positive at x0 and zero outside U

  • ​Then  $\displaystyle F:X \rightarrow \mathbb R^J , x\mapsto (f_\alpha(x))_{\alpha \in J}$ is an inbedding of X in  $\displaystyle F:X \rightarrow \mathbb R^J , x\mapsto (f_\alpha(x))_{\alpha \in J}$

• ​Theorem

  • ​Let (X,d) be a metric space, (X,Td) is the corresponding topology space

  • ​1) A, a subset of X, as a topology subspace of (X,Td) 

  • ​2) A with metric $\displaystyle d|_A:A\times A \rightarrow \mathbb R_+$ 

  • ​1) is same as 2)

• ​Tietze extension theorem

  • ​Let X be a normal space, let A be a closed subspace of X, then we have:

  • ​(1) Any continuous  $\displaystyle f: A \rightarrow [a,b]$ may be extended to a continuous map  $\displaystyle F: X \rightarrow [a,b]$ so that  $\displaystyle f=F|_A$ 

  • ​(2) Any continuous  $f:A \rightarrow \mathbb R$ can be extended to a continuous map  $F:X \rightarrow \mathbb R$ so that  $F|_A=f$ 

• ​Theorem

  • ​Every regular space with a countable basis is normal 

• ​Urysohn metrization theorem

  • ​metrizable : there is a metric d which can generate (X,T) by Td.

  • ​Every regular space with countable basis is metrizable

  • ​claim

    • ​there exists a countable collection of continuous functions  $\displaystyle f_n:X \rightarrow [0,1]$ having the property that :

    • ​Given any point  $\displaystyle x_0 \in X$ and any neighbourhood U of  $\displaystyle x_0$, there exists an index n so that fn is positive at  $\displaystyle x_0$ and zero outside U.

• ​partition of unity

  • ​Def

    • ​Let  $\{u_1,...,u_n\}$ be a finite indexed open cover of X

    • ​An indexed family of continuous maps  $\phi_i :X\rightarrow[0,1]$ is called a partition of unity dominated by  $\{u_i\}$ if:

    • ​(1)  $\text{supp} (\phi_i)\subseteq u_i$ 

    • ​(2)  $\displaystyle \sum_{i=1}^n\phi_i(x)=1, \forall x\in X$ 

  • ​Existence of finite partition of unity

    • ​ any open cover $\displaystyle \{ u_1,...,u_n\}$ of a normal space X can dominate a partition of unity.