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# 数学代写|拓扑学代写TOPOLOGY代考|Homeomorphisms

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## 数学代写|拓扑学代写TOPOLOGY代考|Homeomorphisms

Definition 1.9 A homeomorphism is a continuous and bijective map with continuous inverse. Two subsets in $\mathbb{R}^n$ are called homeomorphic if there is a homeomorphism mapping one to the other.

In the eyes of the topologist homeomorphic spaces are indistinguishable. For instance, he won’t see any difference between the four intervals
$$] 0,1[, \quad] 0,2[, \quad] 0,+\infty[,]-\infty,+\infty[$$

(see Exercise 1.8 for the notation). Indeed, the maps
$$\begin{gathered} f:]-\infty,+\infty[\longrightarrow] 0,+\infty\left[\quad f(x)=e^x,\right. \ g:] 0,+\infty[\longrightarrow] 0,1\left[\quad g(x)=e^{-x},\right. \ h:] 0,1[\longrightarrow] 0,2[\quad h(x)=2 x \end{gathered}$$
are homeomorphisms.
Through a topologist’s spectacles a circle and a square:
$$S^1=\left{(x, y) \in \mathbb{R}^2 \mid x^2+y^2=1\right} \text { and } P=\left{(x, y) \in \mathbb{R}^2|| x|+| y \mid=1\right}$$
are identical. It is easy to see that
$$f: S^1 \rightarrow P, \quad f(x, y)=\left(\frac{x}{|x|+|y|}, \frac{y}{|x|+|y|}\right)$$
and
$$g: P \rightarrow S^1, \quad g(x, y)=\left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)$$
are both continuous and inverse to one another, see Fig. 1.8.

## 数学代写|拓扑学代写TOPOLOGY代考|Notations and Basic Concepts

If $X$ is a set we’ll write $x \in X$ if $x$ belongs to $X$, that is, if $x$ is an element of $X$. We’ll indicate with $\emptyset$ the empty set, while the symbols ${*}$ and ${\infty}$ will both denote the singleton, the set with only one element. A set is called finite if it contains a finite number of elements, and we’ll write $|X|=n$ if $X$ has exactly $n$ elements. A set that isn’t finite is called infinite.

If $A$ and $B$ are sets we write $A \subset B$ if $A$ is contained in $B$, i.e. when every element of $A$ is an element of $B$. We use $A \subset B, A \neq B$, or $A \subsetneq B$, in case $A$ is strictly contained in $B$. The set $A$ is said to meet or intersect $B$ (then $A$ and $B$ meet, or intersect) if their intersection $A \cap B$ isn’t empty.

Example 2.1 A consequence of the definition of $\subset$ is that $\emptyset \subset A$, for any set $A$. Even if this doesn’t convince you completely you must accept it anyhow, at least as a convention. In this way, given any property $\mathfrak{p}$ defined on the elements of $A$, it makes sense to write
$${a \in A \mid \mathfrak{p}(a)} \subset A$$
where ${a \in A \mid \mathfrak{p}(a)}$ denotes the set of elements in $A$ for which $\mathfrak{p}$ is true. Similarly, for any set $A$ there is a unique map $\emptyset \rightarrow A$. Sometimes it may be better to think of the singleton as the set of maps from the empty set to itself.

## 数学代写|拓扑学代写TOPOLOGY代考|Homeomorphisms

1.9同胚是具有连续逆的连续双射映射。如果$\mathbb{R}^n$中的两个子集之间存在同胚映射，则称为同胚子集。

$$] 0,1[, \quad] 0,2[, \quad] 0,+\infty[,]-\infty,+\infty[$$

(见练习1.8中的符号)。事实上，这些地图
$$\begin{gathered} f:]-\infty,+\infty[\longrightarrow] 0,+\infty\left[\quad f(x)=e^x,\right. \ g:] 0,+\infty[\longrightarrow] 0,1\left[\quad g(x)=e^{-x},\right. \ h:] 0,1[\longrightarrow] 0,2[\quad h(x)=2 x \end{gathered}$$

$$S^1=\left{(x, y) \in \mathbb{R}^2 \mid x^2+y^2=1\right} \text { and } P=\left{(x, y) \in \mathbb{R}^2|| x|+| y \mid=1\right}$$

$$f: S^1 \rightarrow P, \quad f(x, y)=\left(\frac{x}{|x|+|y|}, \frac{y}{|x|+|y|}\right)$$

$$g: P \rightarrow S^1, \quad g(x, y)=\left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)$$

## 数学代写|拓扑学代写TOPOLOGY代考|Notations and Basic Concepts

$${a \in A \mid \mathfrak{p}(a)} \subset A$$

## 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。