assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Topological Spaces

In the previous chapter we defined the concept of a continuous mapping of one metric space into another, and this definition was formulated in terms of the metrics on the spaces involved. It often happens, however, that it is convenient-even essential-to be able to speak of continuous mappings in situations where no useful metrics are defined, readily definable, or capable of being defined. In order to deal effectively with circumstances of this kind, it is necessary for us to liberate our concept of continuity from its dependence on metric spaces.

Theorem 13-C shows that the continuity of a mapping of one metric space into another can be expressed solely in terms of open sets, without any direct reference to metrics. This suggests the possibility of discarding metrics altogether and of replacing them as the source of our theory by open sets. With this in mind, our attention is drawn to Theorem 10-D, which gives the main internal properties of the class of open sets in a metric space. These two theorems provide the leading hint on which we base our generalization of metric spaces to topological spaces – a topological space being simply a non-empty set in which there is given a class of subsets, called open sets, with the properties expressed in Theorem 10-D.

Our underlying purpose in this and the next four chapters is to study topological spaces and continuous mappings of topological spaces into one another. We shall see that these spaces provide the ideal context for a theory of continuity in its purest form.

## 数学代写|拓扑学代写Topology代考|THE DEFINITION AND SOME EXAMPLES

Let $X$ be a non-empty set. A class $\mathrm{T}$ of subsets of $X$ is called a topology on $X$ if it satisfies the following two conditions:
(1) the union of every class of sets in $T$ is a set in $T$;
(2) the intersection of every finite class of sets in $\mathrm{T}$ is a set in $\mathrm{T}$.
A topology on $X$ is thus a class of subsets of $X$ which is closed under the formation of arbitrary unions and finite intersections. A topological space consists of two objects: a non-empty set $X$ and a topology $T$ on $X$. The sets in the class $\mathbf{T}$ are called the open sets of the topological space $(X, T)$, and the elements of $X$ are called its points. It is customary to denote the topological space $(X, T)$ by the symbol $X$ which is used for its underlying set of points. No harm can come from this practice if one clearly understands that a topological space is more than merely a non-empty set: it is a non-empty set together with a specific topology on that set. We shall often be considering several topologies on a single given set, and in these circumstances distinct topologies make the set into distinct topological spaces. We observe that the empty set and the full space are always open sets in every topological space, since they are the union and intersection of the empty class of sets, which is a subclass of every topology.

We now list several simple examples of topological spaces. In order to exhibit a topological space, one must specify a non-empty set, tell which subsets are to be considered the open sets, and verify that this given class of sets satisfies conditions (1) and (2) above. In the examples which follow, we leave this third step to the reader.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考| THE DEFINITION AND SOME EXAMPLES

（1） 集合中每个集合类的并集T是设置T;
（2） 每个有限集合类的交集T是设置T.

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师