# 数学代写|实分析作业代写Real analysis代考|Subspaces and Products

#### Doug I. Jones

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## 数学代写|实分析作业代写Real analysis代考|Subspaces and Products

When working with functions on the real line, one frequently has to address situations in which the domain of the function is just an open interval or a closed interval, rather than the whole line. When one uses the $\epsilon-\delta$ definition of continuity, the subject does not become much more cumbersome, but it can become more cumbersome if one uses some other definition, such as one involving limits. The theory of metric spaces has a device for addressing smaller domains than the whole space – the notion of a subspace – and then the theory of functions on a subspace stands on an equal footing with the theory of functions on the whole space.

Let $(X, d)$ be a metric space, and let $A$ be a nonempty subset of $X$. There is a natural way of making $A$ into a metric space, namely by taking the restriction $\left.d\right|_{A \times A}$ as a metric for $A$. When we do so, we speak of $A$ as a subspace of $X$. When there is a need to be more specific, we may say that $A$ is a metric subspace of $X$. If $A$ is an open subset of $X$, we may say that $A$ is an open subspace; if $A$ is a closed subset of $X$, we may say that $A$ is a closed subspace.

Proposition 2.26. If $A$ is a subspace of a metric space $(X, d)$, then the open sets of $A$ are exactly all sets $U \cap A$, where $U$ is open in $X$, and the closed sets of $A$ are all sets $F \cap A$, where $F$ is closed in $X$.

PROOF. The open balls in $A$ are the intersections with $A$ of the open balls of $X$, and the statement about open sets follows by taking unions. The closed sets of $A$ are the complements within $A$ of all the open sets of $A$, thus all sets of the form $A-(U \cap A)$ with $U$ open in $X$. Since $A-(U \cap A)=A \cap U^c$, the statement about closed sets follows.

## 数学代写|实分析作业代写Real analysis代考|Properties of Metric Spaces

This section contains two results about metric spaces. One lists a number of “separation properties” of sets within any metric space. The other concerns the completely different property of “separability,” which is satisfied by some metric spaces and not by others, and it says that separability may be defined in any of three equivalent ways.

Proposition 2.30 (separation properties). Let $(X, d)$ be a metric space. Then
(a) every one-point subset of $X$ is a closed set, i.e., $X$ is $\mathbf{T}_1$,
(b) for any two distinct points $x$ and $y$ of $X$, there are disjoint open sets $U$ and $V$ with $x \in U$ and $y \in V$, i.e., $X$ is Hausdorff,
(c) for any point $x \in X$ and any closed set $F \subseteq X$ with $x \notin F$, there are disjoint open sets $U$ and $V$ with $x \in U$ and $F \subseteq V$, i.e., $X$ is regular,
(d) for any two disjoint closed subsets $E$ and $F$ of $X$, there are disjoint open sets $U$ and $V$ such that $E \subseteq U$ and $F \subseteq V$, i.e., $X$ is normal,
(e) for any two disjoint closed subsets $E$ and $F$ of $X$, there is a continuous function $f: X \rightarrow[0,1]$ such that $f$ is 0 exactly on $E$ and $f$ is 1 exactly on $F$.

PROOF. For (a), the set ${x}$ is the intersection of all closed balls $B(r ; x)$ for $r>0$ and hence is closed by Corollary 2.18 and Proposition $2.8 \mathrm{~b}$. For (e), the function $f(x)=D(x ; E) /(D(x ; E)+D(x ; F))$ is continuous by Proposition 2.16 and Corollary 2.29 and takes on the values 0 and 1 exactly on $E$ and $F$, respectively, by Proposition 2.19 .

For (d), we need only apply (e) and Proposition $2.15 \mathrm{~b}$ with $U=f^{-1}\left(\left(-\infty, \frac{1}{2}\right)\right)$ and $V=f^{-1}\left(\left(\frac{1}{2},+\infty\right)\right)$. Conclusions (a) and (d) imply (c), and conclusions (a) and (c) imply (b). This completes the proof.

A base $\mathcal{B}$ for a metric space $(X, d)$ is a family of open sets such that every open set is a union of members of $\mathcal{B}$. The family of all open balls is an example of a base.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Properties of Metric Spaces

(a) $X$的每一个单点子集都是一个闭集，即$X$为$\mathbf{T}_1$;
(b)对于$X$的任意两个不同的点$x$和$y$，存在与$x \in U$和$y \in V$不相交的开集$U$和$V$，即$X$为Hausdorff;
(c)对于任意点$x \in X$和任意闭集$F \subseteq X$与$x \notin F$，存在不相交的开集$U$和$V$与$x \in U$和$F \subseteq V$，即$X$是正则的;
(d)对于$X$的任意两个不相交的闭子集$E$和$F$，存在不相交的开集$U$和$V$，使得$E \subseteq U$和$F \subseteq V$为正态，即$X$为正态;
(e)对于$X$的任意两个不相交的闭子集$E$和$F$，存在一个连续函数$f: X \rightarrow[0,1]$，使得$f$在$E$上精确为0,$f$在$F$上精确为1。

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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。

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