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# 数学代写|数学分析作业代写Mathematical Analysis代考|Definitions and Basic Properties

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties

While the metric topology is often sufficient for most introductory courses in analysis, a good understanding of the elements of general topology is essential for any advanced study of analysis. An attempt to define topology in a paragraph is quite difficult and not likely to be successful, but we offer the following narrative for the satisfaction of the the reader who insists on an overview of the subject. We saw in chapter 4 that the collection of open sets generated by a metric has many intrinsic properties independent of the defining metric. In this section, we study the arrangement of the collection of open sets, or the topology, in a metric-free context. Every metric space is a topological space; hence all results for topological spaces (which are meaningful in the metric setting) are also valid for metric spaces, but not conversely. We often fall back on the metric case to gain insight into both subjects. We will encounter in this section many of the definitions that appeared in chapter 4, such as closure, interior, and boundary. We include those definitions again in this chapter for ease of reference. However, the proofs that duplicate those in chapter 4 are omitted. The amount of duplication is small and does not rise to the level of redundancy. We encourage the reader to compare results in this section to their counterparts in the previous chapter. The exercise is insightful.
Let $X$ be a nonempty set, and let $\mathcal{T}$ be a collection of subsets of $X ; \mathcal{T}$ is called a topology on $X$ if
(a) $\varnothing$ and $X$ are in $\mathcal{T}$,
(b) the union of an arbitrary family of members of $\mathcal{T}$ is a member of $\mathcal{T}$, and
(c) the intersection of two members of $\mathcal{T}$ is a member of $\mathcal{T}$.
Thus $\mathcal{T}$ is closed under the formation of arbitrary unions and finite intersections. The members of $\mathcal{T}$ are called the open subsets of $X$, and the pair $(X, \mathcal{T})$ is called a topological space.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Subspace Topology

Let $(X, \mathcal{J})$ be a topological space, and let $Y$ be a subset of $X$. Define $\mathcal{T}Y={Y \cap U$ : $U \in \mathcal{T}}$. It is easy to verify that $\mathcal{J}_Y$ is a topology on $Y$. For example, if $\left{Y \cap U\alpha\right}_\alpha$ is a collection of members of $\mathcal{T}Y$, then $\cup\alpha\left(Y \cap U_\alpha\right)=Y \cap\left(\cup_\alpha U_\alpha\right)$, which is in $\mathcal{T}Y$ because $\cup\alpha U_\alpha \in \mathcal{T}$. Verifying that $\mathcal{J}_Y$ is closed under the formation of finite intersections is straightforward.
Definition. The topology $\mathcal{T}_Y$ is known as the relative, subspace, or restricted topology on $Y$ induced by the topology $\mathcal{T}$.
Theorem 5.1.6. Let $A \subseteq Y$, and let $\bar{A}_Y$ denote the closure of $A$ in $\left(Y, \mathcal{T}_Y\right)$. Then $\bar{A}_Y=\bar{A} \cap Y$.
Proof. Since $\bar{A}$ is closed in $X, \bar{A} \cap Y$ is closed in $Y$. Since $A \subseteq \bar{A} \cap Y, \bar{A}_Y \subseteq \bar{A} \cap Y$. We prove the reverse containment. Since $\bar{A}_Y$ is closed in $Y$, there exists a closed subset $F$ of $X$ such that $\bar{A}_Y=F \cap Y$. Thus $F$ is a closed subset of $X$, and $A \subseteq F$. Hence $\bar{A} \subseteq F$, and $\bar{A} \cap Y \subseteq F \cap Y=\bar{A}_Y$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties

(a) $\varnothing$和$X$分别代表$\mathcal{T}$;
(b) $\mathcal{T}$的任意成员族的并集是$\mathcal{T}$的成员，并且
(c) $\mathcal{T}$的两个元素的交集是$\mathcal{T}$的一个元素。

## 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。