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

# 数学代写|拓扑学代写TOPOLOGY代考|MATH625 Spheres as Surfaces

avatest™

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 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。