Posted on Categories:Topology, 拓扑学, 数学代写

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## 数学代写|拓扑学代写TOPOLOGY代考|Spheres as Surfaces

The most familiar example of a surface (other than an open set in $\mathbb{R}^2$ ) is a sphere $\mathbb{S}^2$, since we live on one. If we look around a bit at the surface of our planet, we might be inclined to suspect that Earth is flat, because it appears flat when we can only see a bit of it at a time.

Let us now start a rigorous proof that a sphere is a surface according to our definition. To do so, we need to show that for every point $p \in \mathbb{S}^2$, there is an open set $U$ of $\mathbb{S}^2$ containing $p$, and a homeomorphism $f: U \rightarrow V \subset \mathbb{R}^2$. Thus we must first choose an appropriate open set $U$ for each point $p$, and then construct the required homeomorphism. Note that the latitude and longitude coordinates we introduced above do not yet suffice. There are two reasons: the first is that they are only welldefined on part of $\mathbb{S}^2$, so we would only be able to prove that this part of $\mathbb{S}^2$ is a surface rather than all of $\mathbb{S}^2$; the second is that we have not defined $f$, nor shown the existence of $f^{-1}$, for these coordinates yet. We’ll leave both of these issues for you to ponder on your own, and we will presently prove that $\mathbb{S}^2$ is a surface in a different way.

## 数学代写|拓扑学代写TOPOLOGY代考|Surfaces with Boundary

A natural question prompted by our consideration of hemispheres just now is: What is the nature of the closed hemisphere $\bar{U}{\text {top }}:=\left{(x, y, z) \in \mathbb{S}^2: z \geq 0\right}$ ? Although this object is almost as surface-like as the familiar sphere $\mathbb{S}^2$, we are unfortunately not justified in calling it a surface-at least according to our definition. This is because any point on the boundary of the closed hemisphere, namely any point of the form $(x, y, 0) \in \bar{U}{\text {top }}$, does not satisfy the surface property. For instance, we can form a relatively open set in $\bar{U}{\text {top }}$ containing $(x, y, 0)$ by intersecting $\bar{U}{\text {top }}$ with $B_r((x, y, 0))$. This open set is homeomorphic to a half-disk in $\mathbb{R}^2$ under the projection $f_{\text {top }}$, which is neither open nor closed. This is only one example, but it reflects a general phenomenon: Try as we might, we will never be able to map a relatively open set containing $(x, y, 0)$ to an open set in the plane, because the image of $\bar{U}{\text {top }}$ will always be on only one side of the image of the boundary of $\bar{U}{\text {top }}$.

We would, however, like to include the closed hemisphere $\bar{U}_{\text {top }}$ in our list of allowed “surface-like” objects. Therefore we make a special definition that covers the case of the closed hemisphere and similar surfaces with boundary curves. We’ll need the standard two-dimensional closed half-space defined by $\mathbb{H}^2:={(x, y) \in$ $\left.\mathbb{R}^2: y \geq 0\right}$. We denote its boundary by $\partial \mathbb{H}^2={(x, 0): x \in \mathbb{R}}$.

Definition $2.3$ A surface with boundary $S$ is a non-empty topological space such that for every point $p \in S$, there is an open set $U \subset S$ containing $p$, and a homeomorphism $f: U \rightarrow V$ onto a relatively open subset $V \subset \mathbb{H}^2$.

This definition admits two kinds of points in $S$. There are those points for which the original definition of “surface” holds, namely the homeomorphism $f: U \rightarrow V$ is such that $V$ is contained in the interior of $\mathbb{H}^2$ and is thus an ordinary open set in $\mathbb{R}^2$. And there are those points whose image under $f$ lie on $\partial \mathbb{H}^2$.

## 数学代写|拓扑学代写TOPOLOGY代考|Surfaces with Boundary

\left 缺少或无法识别的分隔符 ？虽然这个物体几乎和我们熟悉的球体一样像表面 $\mathbb{S}^2$, 不幸 的是，我们没有理由将其称为表面 – 至少根据我们的定义。这是因为封闭半球边界上的任意点，即形式的任意 点 $(x, y, 0) \in \bar{U}$ top ，不满足表面性质。例如，我们可以形成一个相对开放的集合 $\bar{U}$ top 含有 $(x, y, 0)$ 通过相 它反映了一个普遍现象: 无论我们怎么努力，我们永远无法映射出一个相对开放的集合，其中包含 $(x, y, 0)$ 到 平面上的开集，因为图像 $\bar{U}$ top 永远只在图像边界的一侧 $\bar{U}$ top .

\right 缺少或无法识别的分隔符 . 我们将其边界表示为 $\partial \mathbb{H}^2=(x, 0): x \in \mathbb{R}$.

## 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。

Posted on Categories:Topology, 拓扑学, 数学代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|拓扑学代写TOPOLOGY代考|Homotopies of Maps and Spaces

In the last chapter, we discussed homotopies of maps between $[0,1]$ and a topological space $X$. We can generalize this to maps between two arbitrary topological spaces $X$ and $Y$. We say that two maps $f, g: X \rightarrow Y$ are homotopic if we can continuously deform one into the other. We can express this notion more formally, in a similar manner to how we defined homotopies of maps between $[0,1]$ and $X$ :

Definition $9.1$ Suppose $X$ and $Y$ are two topological spaces, and $f, g: X \rightarrow Y$ are two continuous maps. Then a homotopy between $f$ and $g$ is a continuous map $H:[0,1] \times X \rightarrow Y$ satisfying the following properties:

$H(0, x)=f(x)$ for all $x \in X$,

$H(1, x)=g(x)$ for all $x \in X$.
If there is a homotopy between $f$ and $g$, then we say that $f$ and $g$ are homotopic. We write $f \sim g$ when $f$ and $g$ are homotopic.

Example Let $X$ be the interval $[0,1]$, and let $Y$ be the single point 0 . Then $X$ and $Y$ are homotopy equivalent. To see this, we need to define maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$. We define $f(x)=0$ for all $x \in X$, and $g(0)=0$ (for the only point 0 in $Y$ ). Then $(g \circ f)(x)=0$ for all $x \in X$. To see that this is homotopic to the identity map $h(x)=x$, we need to construct a homotopy $H:[0,1] \times X \rightarrow$ $X$ between them. Our homotopy will be defined by $H(s, x)=s x$. Then we have $H(0, x)=0=(g \circ f)(x)$, and $H(1, x)=x=h(x)$. So this is a homotopy between $(g \circ f)(x)$ and the identity function on $X$.

Now we have to show that $f \circ g$ is homotopic to the identity function on $Y$. But this is easier, because both functions are the same function that sends the only point in $Y$ to itself. The homotopy $J$ between them is defined by $J(s, x)=0$.

## 数学代写|拓扑学代写TOPOLOGY代考|Computing the Fundamental Group of a Circle

So far, it is not yet clear whether the fundamental group is an interesting invariantthat is, does it ever distinguish spaces? Are there any spaces at all with nontrivial fundamental group? In case the name didn’t give it away, here’s a spoiler: yes! We will show that the circle has nontrivial fundamental group.

Before we do this, let us see intuitively why we ought to believe that the circle has nontrivial fundamental group. Suppose our circle is the set $\mathbb{S}^1=\left{(x, y): x^2+y^2=\right.$ 1} $\subset \mathbb{R}^2$. Let us pick as our basepoint the point $p=(1,0)$. Let us consider the loop $\alpha$ on the circle; $\alpha$ is a map $\alpha:[0,1] \rightarrow \mathbb{S}^1$ so that $\alpha(0)=\alpha(1)=p$, and we will choose it to be the loop $\alpha(t)=(\cos 2 \pi t, \sin 2 \pi t)$, so it is a loop of constant speed that goes around the circle once in the counterclockwise direction.

This loop appears not to be homotopic to the trivial loop: it seems that this loop goes around once, and the trivial loop goes around 0 times. But how can we prove that, by doing some clever homotopy, we can’t shrink it down to a point?

There are several ways of proving this, and the different techniques highlight different properties of fundamental groups. In this section, we’ll see a way to do it using a first example of covering spaces, while in the next chapter we’ll see a different proof. We won’t talk more about covering spaces in general in this book, but the procedure we employ here to compute fundamental groups is very general and can be used to compute the fundamental group of any reasonably nice space.
The outline of the proof is the following: We want to start with a loop on the circle, lift it up to some other space, and see what the lifted version of the loop looks like.

## 数学代写|拓扑学代写TOPOLOGY代考|Homotopies of Maps and Spaces

$H(0, x)=f(x)$ 对全部 $x \in X ，$
$H(1, x)=g(x)$ 对全部 $x \in X$.

$H:[0,1] \times X \rightarrow X$ 它们之间。我们的同伦定义为 $H(s, x)=s x$. 然后我们有 $H(0, x)=0=(g \circ f)(x)$ 和 $H(1, x)=x=h(x)$. 所以这是之间的同伦 $(g \circ f)(x)$ 和身份函数 $X$.

## 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。