Theorem 18.4 in James Munkres “Topology” states that if a function \(f : A \rightarrow X \times Y\) is continuous, its coordinate functions \(f_1 : A \rightarrow X\) and \(f_2 : A \rightarrow Y\) are also continuous, and the converse is also true. Th…

Exercise 22.3 Let \(\pi_1: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be projection on the first coordinate. Let \(A\) be the subspace of \(\mathbb{R}\times\mathbb{R}\) consisting of all points \(x \times y\) for which either \(x \geq 0\)…

Exercise 22.6 Recall that \(\mathbb{R}_{K}\) denotes the real line in the \(K\)-topology. Let \(Y\) be the quotient space obtained from \(\mathbb{R}_K\) by collapsing the set \(K\) to a point; let \(p: \mathbb{R}_K \rightarrow Y\) be the quotient map…

Exercise 1. Let \(X\) be a space. Let \(\mathcal{D}\) be a collection of subsets of \(X\) that is maximal with respect to the finite intersection property (FIP). (a) Show that \(x \in \bar{D}\) for every \(D \in \mathcal{D}\) if and only if every nei…

Example 1 Let \(X\) be the subspace \([0,1]\cup[2,3]\) of \(\mathbb{R}\), and let \(Y\) be the subspace \([0,2]\) of \(\mathbb{R}\). The map \(p: X \rightarrow Y\) defined by \[ p(x)=\begin{cases} x & \text{for}\; x \in [0,1],\\ x-1 & \text{for}\;…

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…

Lemma 21.2 (The sequence lemma) Let \(X\) be a topological space; let \(A \subset X\). If there is a sequence of points of \(A\) converging to \(x\), then \(x \in \bar{A}\); the converse holds if \(X\) is metrizable. Proof a) Sequence convergence \(\…

Proof of Theorem 20.3 Theorem 20.3 The topologies on \(\mathbb{R}^n\) induced by the euclidean metric \(d\) and the square metric \(\rho\) are the same as the product topology on \(\mathbb{R}^n\). Proof: a) Prove the two metrics can mutually limit ea…

Theorem 20.4 The uniform topology on \(\mathbb{R}^J\) is finer than the product topology and coarser than the box topology; these three topologies are all different if \(J\) is infinite. Proof: a) Prove the uniform topology is finer than the product…

Theorem 16.3 If \(A\) is a subspace of \(X\) and \(B\) is a subspace of \(Y\), then the product topology on \(A \times B\) is the same as the topology \(A \times B\) inherits as a subspace of \(X \times Y\). Comment: To prove the identity of two topo…

许久以前,我读到了侯捷先生于<深入浅出MFC>一书中所写的“勿在浮砂筑高台”这句话,颇受警醒与启发.如今在工科领域已摸索多年,亦逐渐真切而深刻地认识到,若没有坚实.完整.细致的数学理论作为基石,任何技术工作与所谓的“唯象”理论研究也都无异于浮沙筑台,经不起举一反三的推敲与寻根究底的质疑.而这也像是不系安全带的无照司机驾车在高速公路上疾驰,终究逃不掉翻到沟里的命运.拓扑学作为数学理论大厦的一块基石,既起到了稳固上层建筑的作用,又提供了不同于我们通常的眼光和感性直观的视角与思路,值得细心学习和反复…

tec 客户端的接受代码 import socket def main(): # 1. 创建tcp的套接字 tcp_socket = socket.socket(socket.AF_INET, socket.SOCK_STREAM) # 2. 链接服务器 # tcp_socket.connect(("192.168.33.11", 7890)) server_ip = input("请输入要链接的服务器的ip:") server_port = int(input(&…

part 组员                工作              工作耗时/h 明日计划 工作耗时/h    UI 冯晓云    完成主页面设计和非功能性PDF reader UI设计实现    8  修补弹窗UI实现   6 foxit PDF Reader 朱玉影 基本完成pdf文本内搜索部分代码,包括当前页面某一结果的显示,下一结果,前一结果 的显示,以及得到所有搜索结果  8  修改搜索部分代码,使其更加robust  4 login 周以舟         async 林建…

Arithmetic operations taught in elementary schools are continuous in the high level topological point of view. This signifies that there is literally no clear boundary between simple and complex, low and high concepts. Instead, they both play indispe…

This post summarises different ways of constructing continuous functions, which are introduced in Section 18 of James Munkres "Topology". Constant function. Inclusion function. N.B. The function domain should have the subspace topology relative…

In James Munkres "Topology" Section 22, the quotient space is defined as below. Definition Let \(X\) be a topological space, and let \(X^*\) be a partition of \(X\) into disjoint subsets whose union is \(X\). Let \(p: X \rightarrow X^*\) be the…

In James Munkres "Topology", the concept for a tuple, which can be \(m\)-tuple, \(\omega\)-tuple or \(J\)-tuple, is defined from a function point of view as below. Let \(X\) be a set. Let \(m\) be a positive integer and \(\{ 1, \cdots, m \}\) be…

According to Wikipedia, the well known barber paradox states like this: The barber is the "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? Actually, this paradox is directly…

一.概述 sqoop 是 apache 旗下一款“Hadoop 和关系数据库服务器之间传送数据”的工具. 核心的功能有两个: 导入.迁入 导出.迁出 导入数据:MySQL,Oracle 导入数据到 Hadoop 的 HDFS.HIVE.HBASE 等数据存储系统 导出数据:从 Hadoop 的文件系统中导出数据到关系数据库 mysql 等 Sqoop 的本质还是一个命令行工具,和 HDFS,Hive 相比,并没有什么高深的理论. sqoop: 工具:本质就是迁移数据, 迁移的方式:就是把sqoo…

来源https://www.cnblogs.com/qingyunzong/p/8807252.html 一.概述 sqoop 是 apache 旗下一款“Hadoop 和关系数据库服务器之间传送数据”的工具. 核心的功能有两个: 导入.迁入 导出.迁出 导入数据:MySQL,Oracle 导入数据到 Hadoop 的 HDFS.HIVE.HBASE 等数据存储系统 导出数据:从 Hadoop 的文件系统中导出数据到关系数据库 mysql 等 Sqoop 的本质还是一个命令行工具,和 HDFS,…

18.1.[root@web01 nginx]# mysql -uroot -p123456 mysql> show databases; #显示所有的数据库: +--------------------+ | Database | +--------------------+ | information_schema | | mysql | | performance_schema | | test | +--------------------+ 4 rows in set (0.00 se…

系列目录 这一节,我们插入数据来看看数据流,让各位同学,知道这个权限表交互是怎么一个流程,免得大家后天雾里来雾里去首先我再解释一些表,SysUser和SysRole表不用解释了. SysRoleSysUser:这个是角色和用户的对应表 一个角色可以对应多个用户,一个用户可以对应多个角色 SysModuleOperate:模块的操作码表,我把每个Action都看作是一个操作码,或者每个方法可以看作是一个操作码 SysRight:这个是角色和模块的关系表,只有被模块授权的角色才能设置权限 SysRi…

原文:构建ASP.NET MVC4+EF5+EasyUI+Unity2.x注入的后台管理系统(18)-权限管理系统-表数据 这一节,我们插入数据来看看数据流,让各位同学,知道这个权限表交互是怎么一个流程,免得大家后天雾里来雾里去首先我再解释一些表,SysUser和SysRole表不用解释了. SysRoleSysUser:这个是角色和用户的对应表 一个角色可以对应多个用户,一个用户可以对应多个角色 SysModuleOperate:模块的操作码表,我把每个Action都看作是一个操作码,或者每个…

注:本次成绩排行榜是针对结对项目二的点评分数 作业要求 http://www.cnblogs.com/easteast/p/7604534.html 评分细则 (1)有贴生成得最"好"的数据链接(5'),并详细解释了"数据生成"程序的原理与所考虑的因素.(10') (2)数据建模及匹配程序的思路及实现方式方面,给分点主要在几个方面:是否有清晰明了的框架图或结构图(10'),算法的理论基础或源自的相似问题,对原始算法做出的改进等(10'),建模时考虑了哪些因素,是如何…

1.环境准备 centos7 .docker-ce18.06.1-ce.kubeadm.kubelet.kubectl 2.安装 yum安装,准备repo文件 docker: [docker-ce-stable] name=Docker CE Stable - $basearch baseurl=https://download.docker.com/linux/centos/7/$basearch/stable enabled= gpgcheck= gpgkey=https://downloa…

FROM: http://www.tecmint.com/18-tar-command-examples-in-linux/ 18 Tar Command Examples in Linux By Ravi Saive Under: CentOS, Fedora, Linux Commands, Linux Distros, RedHat On: September 15, 2012 The Linux “tar” stands for tape archive, which is used b…

ODA: After Apply ODA 12.2.1.2.0 Patch, Unable to Create TableSpace Due to [ORA-15001: diskgroup "DATA" does not exist or is not mounted | ORA-15040: diskgroup is incomplete] (Doc ID 2375553.1) APPLIES TO: Oracle Database Appliance Software - Ver…

Header 队名:To Be Done 组长博客 作业博客 团队项目进行情况 燃尽图(组内共享) 展示Git当日代码/文档签入记录(组内共享) 注: 由于GitHub的免费范围内对多人开发存在较多限制,因此我们不会使用GitHub作为主代码仓库,而是使用自建的Git基础设施,我们可能不定期将代码镜像到GitHub 组员情况 王永福(组长) 过去两天完成了哪些任务 完成了内存泄露问题的修复,测试确实已修复 用户信息界面绘制 甚至自己造了个缓存View的Fragment基类轮子 接下来的计划 继续…

Header 队名:To Be Done 组长博客 作业博客 团队项目进行情况 燃尽图(组内共享) 展示Git当日代码/文档签入记录(组内共享) 注: 由于GitHub的免费范围内对多人开发存在较多限制,因此我们不会使用GitHub作为主代码仓库,而是使用自建的Git基础设施,我们可能不定期将代码镜像到GitHub 组员情况 王永福(组长) 过去两天完成了哪些任务 完成了内存泄露问题的修复,测试确实已修复 用户信息界面绘制 甚至自己造了个缓存View的Fragment基类轮子 接下来的计划 继续…

Header 队名:To Be Done 组长博客 作业博客 团队项目进行情况 燃尽图(组内共享) 展示Git当日代码/文档签入记录(组内共享) 注: 由于GitHub的免费范围内对多人开发存在较多限制,因此我们不会使用GitHub作为主代码仓库,而是使用自建的Git基础设施,我们可能不定期将代码镜像到GitHub 组员情况 王永福(组长) 过去两天完成了哪些任务 完成了内存泄露问题的修复,测试确实已修复 用户信息界面改进绘制 接下来的计划 继续完善轨迹记录 完成用户信息接口 应用后台定位和保活…