连续映射

前置

• 子空间

定义

• 称$f:(X,{\tau }_X)\to (Y,{\tau }_Y)$为连续映射(continuous map)，若对$Y$的任意开集$V$，$f^{-1}(V)$是$X$的开集
• 称$X$与$Y$同胚(homeomorphic)，若存在双连续函数$f$使得$Y=f[X]$，双连续函数是指$f$是双射、连续、且$f^{-1}$连续。
• 称$f:X\to Y$为局部同胚(local homeomorphism)，若对于任意$x\in X$，存在$x$的开邻域$U$，$f(U)$为$Y$的开集，且$f{|}_U$为同胚映射
• 称$f:X\to Y$为拓扑嵌入(topological embedding)， 若 $f$为连续单射且$f$为$X$到$f[X]$的同胚映射
• 称$f:X\to Y$为拓扑浸入(topological immersion)，若对任一$x\in X$,都存在$N\in {\mathcal{N}}_x$,使得$f{|}_N$为拓扑嵌入

性质

1. $f:X\to Y$是连续映射 $⟺$ $\forall N\in {\mathcal{N}}_{f(x)},f^{-1}(N)\in {\mathcal{N}}_x$

2. 设$(X,\mathcal{U}),(Y,\mathcal{V})$为两个拓扑空间，$\mathcal{S},\mathcal{B}$分别为$Y$的子基和基，$f:X\to Y$,则以下个条件等价：

   • $f$连续
   • $\forall V\in \mathcal{V},f^{-1}(V)\in \mathcal{U}$
   • $\forall S\in \mathcal{S},f^{-1}(S)\in \mathcal{U}$
   • $\forall B\in \mathcal{B},f^{-1}(B)\in \mathcal{U}$

3. $f:X\to Y$连续 $\Rightarrow$ 对于任意序列$\{x_n{\}}_{n=1}^{\infty }$，若$\{x_n{\}}_{n=1}^{\infty }$收敛到$x$，则$\{f(x_n){\}}_{n=1}^{\infty }$收敛到$f(x)$

4. 连续映射复合连续映射为连续映射

5. 坐标映射${\pi }_t:Y={\prod }_{t\in T}{Y}_t\to Y_t$为连续映射

6. $f:(X,\tau )\to ({\prod }_{t\in T}{Y}_t,{\prod }_{t\in T}{\tau }_t)$为连续映射 $⟺$ 对任意$t\in T$，$f_t={\pi }_t\circ f:(X,\tau )\to (Y_t,{\tau }_t)$为连续映射

7. 对于双射$f:X\to Y$，以下三个命题等价：1) $f$为同胚；2) $f$为连续开映射；3) $f$为连续闭映射

8. 若$X$为第一可数空间，$f:X\to Y$为连续映射 $⟺$ 对于$X$中的序列$\{x_n{\}}_{n=1}^{\infty }$，若$\{x_n{\}}_{n=1}^{\infty }$收敛到$x$，则$\{f(x_n){\}}_{n=1}^{\infty }$收敛到$f(x)$

9. $X,Y$为拓扑空间，且$Y$为 Hausdorff 空间，$f,g:X\to Y$为连续映射，则$A=\{x\in X\mid f(x)=g(x)\}$为闭集

10. $X,Y$为拓扑空间，且$Y$为 Hausdorff 空间，$f:D\to Y$为连续映射,$D$ 是$X$的稠密子集，则$f$唯一延拓到$X$上

11. Continuity Is Local： Continuity is a local property, in the following sense:
    if $F:X\to Y$ is a map between topological spaces such that every point $p\in X$ has a open neighborhood $U$ on which the restriction $F{|}_U$ is continuous, then $F$ is continuous.

12. Gluing Lemma for Continuous Maps
    Let $X$ and $Y$ be topological spaces, and suppose one of the following conditions holds:

    • $B_1,\dots ,B_n$ are finitely many closed subsets of $X$ whose union is $X$.
    • $\{B_i{\}}_{i\in A}$ is a collection of open subsets of $X$ whose union is $X$.

    Suppose that for all $i$ we are given continuous maps $F_i:B_i\to Y$ that agree on overlaps:$F_i{|}_{B_i\cap B_j}=F_j{|}_{B_i\cap B_j}.$
    Then there exists a unique continuous map $F:X\to Y$ whose restriction to each $B_i$ is equal to $F_i$.

13. 同胚映射将开集映射为开集

14. 同胚映射将闭集映射到闭集

15. 同胚映射将收敛列映射到收敛列

16. 同胚映射将第一可数空间映射到第一可数空间

17. 同胚映射将第二可数空间映射到第二可数空间

18. 同胚映射将$T_1$空间映射到$T_1$空间

19. 同胚映射将$T_2$空间映射到$T_2$空间

20. 同胚映射将可分空间映射到可分空间

21. 同胚映射将基映射到基

22. 同胚映射保持闭包，$f(\overline{A})=\overline{f(A)}$

23. $X_0\subseteq X$,$f:X\to Y$连续，则$f:X_0\to f(X_0)$也连续,这里的连续是将$X_0$和$f(X_0)$看作$X$和$Y$的子空间

24. Let $X$ be a topological space and let $S$ be a subspace of $X$.

    • CHARACTERISTIC PROPERTY: If $Y$ is a topological space, a map $F:Y\to S$ is continuous if and only if the composition ${\iota }_S\circ F:Y\to X$ is continuous, where ${\iota }_S:S\to X$ is the inclusion map (the restriction of the identity map of $X$ to $S$).
    • The subspace topology is the unique topology on $S$ for which
      the characteristic property holds.
    • The inclusion map ${\iota }_S:S\to X$ is a topological embedding.