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# 数学代写|拓扑学代写Topology代考|MAST31023 COMPACT SPACES

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## 数学代写|拓扑学代写Topology代考|COMPACT SPACES

Iet $X$ be a topological space. A class $\left{G_{i}\right}$ of open subsets of $X$ is said to be an open cover of $X$ if each point in $X$ belongs to at least one $G_{i}$, that is, if $\bigcup_{i} G_{i}=X$. A subclass of an open cover which is itself an open cover is called a subcover. A compact space is a topological space in which every open cover has a finite subcover. A compact subspace of a topological space is a subspace which is compact as a topological space in its own right. We begin by proving two simple but widely used theorems.
Theorem A. Any closed subspace of a compact space is compact. PRoof. Let $Y$ be a closed subspace of a compact space $X$, and let $\left{G_{i}\right}$ be an open cover of $Y$. Each $G_{i}$, being open in the relative topology on $Y$, is the intersection with $Y$ of an open subset $H_{i}$ of $X$. Since $Y$ is closed, the class composed of $Y^{\prime}$ and all the $H_{i}$ ‘s is an open cover of $X$, and since $X$ is compact, this open cover has a finite subcover. If $Y^{\prime}$ occurs in this subcover, we discard it. What remains is a finite class of $H_{i}^{\prime}$ s whose union contains $X$. Our conclusion that $Y$ is compact now follows from the fact that the corresponding $G_{i}$ ‘s form a finite subcover of the original open cover of $Y$.
Theorem B. Any continuous image of a compact space is compact.
PRooF. Let $f: X \rightarrow Y$ be a continuous mapping of a compact space $X$ into an arbitrary topological space $Y$. We must show that $f(X)$ is a compact subspace of $Y$. Let $\left{G_{i}\right}$ be an open cover of $f(X)$. As in the above proof, each $G_{\iota}$ is the intersection with $f(X)$ of an open subset $H_{i}$ of $Y$. It is clear that $\left{f^{-1}\left(H_{i}\right)\right}$ is an open cover of $X$, and by the compactness of $X$ it has a finite subcover. The union of the finite class of $H_{i}$ ‘s of which these are the inverse images clearly contains $f(X)$, so the class of corresponding $G_{i}$ ‘s is a finite subcover of the original open cover of $f(X)$, and $f(X)$ is compact.

It is sometimes quite difficult to prove that a given topological space is compact by appealing directly to the definition. The following theorems give several equivalent forms of compactness which are of ten easier to apply.

Theorem C. A topological space is compact $\Leftrightarrow$ every class of closed sets with empty intersection has a finite subclass with empty intersection.

PRoof. This is a direct consequence of the fact that a class of open sets is an open cover $\Leftrightarrow$ the class of all their complements has empty intersection.

## 数学代写|拓扑学代写Topology代考|PRODUCTS OF SPACES

There are two main techniques for making new topological spaces out of old ones. The first of these, and the simplest, is to form subspaces of some given space. The second is to multiply together a number of given spaces. Our purpose in this section is to describe the way in which the latter process is carried out.

In Sec. 4 we defined what is meant by the product $P_{i} X_{i}$ of an arbitrary non-empty class of sets. We also defined the projection $p_{i}$ of this product onto its $i$ th coordinate set $X_{i}$. The reader should make certain that these concepts are firmly in mind. If each coordinate set is a topological space, then there is a standard method of defining a topology on the product. It is difficult to exaggerate the importance of this definition, and we examine it with great care in the following discussion.
Let us begin by recalling the discussion in Sec. 18 of open rectangles and open strips in the Euclidean plane $R^{2}$. We observed there that the open rectangles form an open base for the topology of $R^{2}$, and also that the open strips form an open subbase for this topology whose generated open base consists of all open rectangles, all open strips, the empty set, and the full space. The topology of the Euclidean plane is of course defined in terms of a metric. If we wish, however, we can ignore this fact and regard the topology of $R^{2}$ as generated in the sense of Theorem 18-D by the class of all open strips. This situation provides the motivation for the more general ideas we now develop.

## 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。