Theorem 19.6 Let \(f: A \rightarrow \prod_{\alpha \in J} X_{\alpha}\) be given by the equation

\[
f(a) = (f_{\alpha}(a))_{\alpha \in J},
\]

where \(f_{\alpha}: A \rightarrow X_{\alpha}\) for each \(\alpha\). Let \(\prod X_{\alpha}\) have the product topology. Then the function \(f\) is continuous if and only if each function \(f_{\alpha}\) is continuous.

Comment: This is an extension of Theorem 18.4, where only two component spaces are involved.

Proof: a) First, we prove the projection map is continuous, which is defined on the Cartesian space constructed from a \(J\)-tuple of component spaces .

For all \(\beta \in J\), let \(\pi_{\beta}: \prod X_{\alpha} \rightarrow X_{\beta}\) be the projection map. For arbitrary open set \(V_{\beta}\) in \(X_{\beta}\), its pre-image under \(\pi_{\beta}\) is \(\pi_{\beta}^{-1}(V_{\beta})\), which is a subbasis element for the product topology on \(\prod X_{\alpha}\). Therefore, \(\pi_{\beta}^{-1}(V_{\beta})\) is open and the projection map \(\pi_{\beta}\) is continuous.

Next, we notice that for all \(\alpha \in J\), the coordinate function \(f_{\alpha}: A \rightarrow X_{\alpha}\) is a composition of the two continuous functions \(f\) and \(\pi_{\alpha}\), i.e. \(f_{\alpha} = \pi_{\alpha} \circ f\). Then according to Theorem 18.2 (c), \(f_{\alpha}\) is continuous.

Remark: Because the box topology is finer than the product topology, the projection map is also continuous when the box topology is adopted for \(\prod X_{\alpha}\). Therefore, this part of the theorem is true for both product topology and box topology.

b) Analysis: To prove the continuity of a function, showing that the pre-image of any subbasis element in the range space is open in the domain space is more efficient than using basis element or raw open set in the range space. In addition, the subbasis element for the product topology on \(\prod X_{\alpha}\) has the form \(\pi_{\beta}^{-1}(U_{\beta})\) with \(U_{\beta}\) being a single coordinate component and open in \(X_{\beta}\). This is the clue of the proof.

For all \(\beta \in J\) and arbitrary open set \(U_{\beta}\) in \(X_{\beta}\), we have \(f_{\beta}^{-1}(U_{\beta}) = f^{-1} \circ \pi_{\beta}^{-1}(U_{\beta})\). Because \(f_{\beta}\) is continuous and \(U_{\beta}\) is open, \(f_{\beta}^{-1}(U_{\beta})\) is open. In addition, \(\pi_{\beta}^{-1}(U_{\beta})\) is an arbitrary subbasis element for \(\prod X_{\alpha}\) with the product topology, whose pre-image under \(f\) is just the open set \(f_{\beta}^{-1}(U_{\beta})\), therefore \(f\) is continuous.

Remark: Part b) of this theorem really depends on the adopted topology for \(\prod X_{\alpha}\), which can be understood as below.

At first, we will show that for all \(\vect{U} = \prod U_{\alpha}\) being a subset of \(\prod X_{\alpha}\), \(f^{-1}(\vect{U}) = \bigcap_{\alpha \in J} f_{\alpha}^{-1}(U_{\alpha})\).

For all \(x \in f^{-1}(\vect{U})\), because \(f(x) \in \vect{U}\), then for all \(\alpha \in J\), \(f_{\alpha}(x) \in U_{\alpha}\), hence \(x \in \bigcap_{\alpha \in J} f_{\alpha}^{-1}(U_{\alpha})\) and \(f^{-1}(\vect{U}) \subset \bigcap_{\alpha \in J} f_{\alpha}^{-1}(U_{\alpha})\).

On the other hand, for all \(x \in \bigcap_{\alpha \in J} f_{\alpha}^{-1}(U_{\alpha})\), we have for all \(\alpha \in J\), \(f_{\alpha}(x) \in U_{\alpha}\). Therefore, \(f(x) \in \vect{U}\) and \(x \in f^{-1}(\vect{U})\). Hence \(\bigcap_{\alpha \in J} f_{\alpha}^{-1}(U_{\alpha}) \subset f^{-1}(\vect{U})\).

Next, if we assign the product topology to \(\prod X_{\alpha}\), for any \(\vect{U} = \prod U_{\alpha}\) with \(U_{\alpha}\) open in \(X_{\alpha}\) and only a finite number of them not equal to \(X_{\alpha}\), it is a basis element of the product topology. Let the set of all indices with which \(U_{\alpha} \neq X_{\alpha}\) be \(\{\alpha_1, \cdots, \alpha_n\}\) and also notice that when \(U_{\alpha} = X_{\alpha}\), \(f_{\alpha}^{-1}(U_{\alpha}) = A\), we have

\[
f^{-1}(\vect{U}) = \bigcap_{\alpha \in J} f_{\alpha}^{-1}(U_{\alpha}) = \bigcap_{i=1}^n f_{\alpha_i}^{-1}(U_{\alpha_i}),
\tag{*}
\label{eq:intersection}
\]

where those \(f_{\alpha}^{-1}(U_{\alpha})\) with \(\alpha \notin \{\alpha_1, \cdots, \alpha_n\}\) do not contribute to the intersection. This indicates that \(f^{-1}(\vect{U})\) is a finite intersection of open sets which is still open. Hence \(f\) is continuous.

However, if the box topology is adopted for \(\prod X_{\alpha}\), qualitatively speaking, because the topology for the range space becomes finer, according to our previous post, it makes a function to be continuous more difficult. Specifically in this theorem, \(f^{-1}(\vect{U})\) in \eqref{eq:intersection} can be an intersection of infinite number of open sets \(U_{\alpha}\) not equal to \(X_{\alpha}\). Thus \(f^{-1}(\vect{U})\) may not be open anymore.

After understanding this point, it is not difficult to construct a counter example for part b) as below.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}\) be defined as \(f(t) = (t, t, \cdots)\). Select a basis element \(\vect{U}\) in \(\mathbb{R}^{\omega}\) such that the intersection of all its coordinate components is not open. For example, \(\vect{U} = \prod_{n=1}^{\infty} (-\frac{1}{n}, \frac{1}{n})\), which is a neighborhood of \(f(0) = (0, 0, \cdots)\).

For any basis element \((a, b)\) in \(\mathbb{R}\) containing \(0\), with \(a < 0\) and \(b > 0\), by letting \(\delta = \min\{-a, b\}\), we have \((-\delta, \delta) \subset (a, b)\) and \(0 \in (-\delta, \delta)\). The image of \((-\delta, \delta)\) under \(f\) is \(\prod_{n=1}^{\infty} (-\delta, \delta)\). Then there exist an \(n_0 \in \mathbb{Z}_+\) such that \((-\delta, \delta)\) is not contained in \((-\frac{1}{n_0}, \frac{1}{n_0})\). Therefore, \(\pi_{n_0}(f((-\delta, \delta)))\) is not contained in \(\pi_{n_0}(\vect{U})\) and \(\pi_{n_0}(f((a, b)))\) is not contained in \(\pi_{n_0}(\vect{U})\). Hence the image of \((a, b)\) under \(f\) is not contained in \(\vect{U}\). This contradicts Theorem 18.1 (4) and \(f\) is not continuous.

James Munkres Topology: Theorem 19.6的更多相关文章

  1. James Munkres Topology: Theorem 20.4

    Theorem 20.4 The uniform topology on \(\mathbb{R}^J\) is finer than the product topology and coarser ...

  2. James Munkres Topology: Theorem 20.3 and metric equivalence

    Proof of Theorem 20.3 Theorem 20.3 The topologies on \(\mathbb{R}^n\) induced by the euclidean metri ...

  3. James Munkres Topology: Theorem 16.3

    Theorem 16.3 If \(A\) is a subspace of \(X\) and \(B\) is a subspace of \(Y\), then the product topo ...

  4. James Munkres Topology: Sec 18 Exer 12

    Theorem 18.4 in James Munkres “Topology” states that if a function \(f : A \rightarrow X \times Y\) ...

  5. James Munkres Topology: Sec 22 Exer 6

    Exercise 22.6 Recall that \(\mathbb{R}_{K}\) denotes the real line in the \(K\)-topology. Let \(Y\) ...

  6. James Munkres Topology: Sec 22 Exer 3

    Exercise 22.3 Let \(\pi_1: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be projection on th ...

  7. James Munkres Topology: Lemma 21.2 The sequence lemma

    Lemma 21.2 (The sequence lemma) Let \(X\) be a topological space; let \(A \subset X\). If there is a ...

  8. James Munkres Topology: Sec 37 Exer 1

    Exercise 1. Let \(X\) be a space. Let \(\mathcal{D}\) be a collection of subsets of \(X\) that is ma ...

  9. James Munkres Topology: Sec 22 Example 1

    Example 1 Let \(X\) be the subspace \([0,1]\cup[2,3]\) of \(\mathbb{R}\), and let \(Y\) be the subsp ...

随机推荐

  1. GOOGLE RANKBRAIN 完整指南

    [译]GOOGLE RANKBRAIN 完整指南 ( 2018 最新版 ) 2018.01.29    来源  http://www.zhidaow.com/post/google-rankbrain ...

  2. FastDFS分布式文件系统客户端安装

    软件安装前提:服务器已配置好LNMP环境安装libfastcommon见FastDFS服务器安装文档(http://www.cnblogs.com/Mrhuangrui/p/8316481.html) ...

  3. [python]python3.7中文手册

    https://pythoncaff.com/docs/tutorial/3.7.0

  4. Lua语法基础(一)

    1. 注释 -- 单行注释 --[[ 多行注释 --]] 2. 运行方式     (1)交互式运行         命令行下 lua进入交互模式     (2)命令行运行         lua + ...

  5. Redis系列八:redis主从复制和哨兵

    一.Redis主从复制 主从复制:主节点负责写数据,从节点负责读数据,主节点定期把数据同步到从节点保证数据的一致性 1. 主从复制的相关操作 a,配置主从复制方式一.新增redis6380.conf, ...

  6. C++: sprintf浮点数精度控制;

    错误的写法: char buf[100]; int num = 10; sprintf(buf, "%.2f", num); ///这种做法是不对的, 按照压栈顺序, 在压入num ...

  7. MSSQL Server2012备份所有数据库到网络共享盘上面,并自动删除几天前的备份。。

    --要备份到哪一服务的IP网络位置,要提前打开文件夹共享.这里还要输入用户名和密码,下面这一行是建立共享 exec master..xp_cmdshell 'net use \\192.168.8.1 ...

  8. sprin源码解析之属性编辑器propertyEditor

    目录 异常信息 造成此异常的原因 bean 配置文件 调用代码 特别说明: 异常解决 注册springt自带的属性编辑器 CustomDateEditor 控制台输出 属性编辑器是何时并如何被注册到s ...

  9. 第十二节:MVC中的一些特殊优化

    一. 删除WebForm视图引擎 在MVC框架中检索视图的顺序为:当前控制器下对应的文件夹的aspx文件→share文件夹aspx文件→当前控制器下对应文件夹的cshtml文件→share文件夹的cs ...

  10. 常用的消息队列中间件mq对比

    原文地址:https://blog.csdn.net/qq_30764991/article/details/80239076 消息队列中间件是分布式系统中重要的组件,主要解决应用耦合,异步消息,流量 ...