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

# 数学代写|拓扑学代写TOPOLOGY代考|MATH374 Identification Spaces

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## 数学代写|拓扑学代写TOPOLOGY代考|Identification Spaces

Now that we have defined the Euler characteristic and seen that it is an invariant of a surface, we would like to be able to calculate it for different types of surfaces. Of course, we know already that $\chi\left(\mathbb{S}^2\right)=2$. At the moment, it isn’t very easy to go beyond this, because it is quite hard to visualize a triangulation of, say, a torus and count the vertices, edges, and faces. (But try and see if you can do it!)

In order to address this limitation, we will develop an easier way to work with surfaces-essentially by drawing them in the plane! To illustrate what we have in mind, let us use the torus as an example. The torus is a surface, so we know that every point of the torus has a small neighborhood that is homeomorphic to part of the plane (the definition of a surface!). But how do we get the whole torus to be part of the plane? We can do this by cheating a bit-but the “cheat” we’ll use will actually end up being a rigorous mathematical operation. First, cut the torus along a circle, as in Figure 3.7. Once we have made this cut, we stretch out the cut torus into a cylinder. Then we make a cut along a line connecting the top and bottom of the cylinder and unroll it. The result is a rectangle, as we can see from looking at Figure 3.8.

We can go the other way too. If we start with a rectangle, we can glue one pair of opposite sides together to create a cylinder, and then we can glue the top and bottom of the cylinder to create a torus. In other words, we get a torus by gluing pairs of opposite sides of a rectangle.

## 数学代写|拓扑学代写TOPOLOGY代考|ID Spaces as Surfaces

Two important questions that we must address are: (1) given a compact surface $S$, can we always obtain an ID space representation for it; (2) supposing we have an ID space as defined in Definition 3.10, how can we tell whether it is an ID space for a surface or a surface with boundary? The answer of the first question is yes-this involves systematically cutting $S$ into triangular faces and assembling these into an ID space. We’ll see how this is done in the next chapter. The second question is interesting to ponder, because we have just seen some examples of ID spaces that are not surfaces amongst the examples above.

Here’s another reason why this is an interesting question. Consider the ID space for the torus that we constructed earlier, namely the square with opposite sides glued together as shown in Figure 3.9. Call it $S$. We can show that only part of the definition of a surface is satisfied. That is, we can show that for every $p \in S$ there is an open set $U$ containing $p$ that can be mapped homeomorphically to an open set in the plane. To see this, consider the following three cases.
(1) If $p$ belongs to the interior of $S$, then the condition is trivially satisfied.
(2) If $p$ belongs to an edge of $S$ but is not a corner of $S$, then we define an open neighborhood of $p$ to be the union of the open half-disk containing $p$ and the open half-disk containing the point $p^{\prime}$ on the opposite edge that is meant to be glued to $p$. Note that, as far as the topology of the torus is concerned, this union of two open half-disks is identical to the open disk from Case (1) because of the gluing instructions that come with $S$. Thus we can also easily map the glued union of open half-disks to the plane.
(3) If $p$ is a corner of $S$, then we define an open neighborhood of $p$ to be the union of four open quarter-disks at the four corners of $S$. Note that, as far as the topology of the torus is concerned, this union of open quarter-disks is identical to the open disk from Case (1) because of the gluing instructions that come with $S$. Thus we can also easily map the glued union of open quarter-disks to the plane.

## 数学代写|拓扑学代写TOPOLOGY代考|识别空间

(1) 如果$p$属于$S$的内部，那么该条件显然是满足的。
(2) 如果$p$属于$S$的一条边，但不是$S$的一个角，那么我们把$p$的一个开放邻域定义为包含$p$的开放半盘和包含对面边上的点$p^{prime}$的开放半盘的联合，而这个点是要与$p$粘连的。请注意，就环状体的拓扑结构而言，由于$S$附带的胶合指令，这个两个开放半盘的联合体与案例（1）中的开放盘是相同的。因此我们也可以很容易地将开放半盘的胶合联盟映射到平面上。
(3) 如果$p$是$S$的一个角，那么我们定义$p$的一个开放邻域是$S$四个角的四个开放四分之一盘的联合。请注意，就环形的拓扑结构而言，由于$S$附带的胶合指令，这个开放四分之一盘的联合体与案例(1)中的开放盘是相同的。因此，我们也可以很容易地将开放四分之一圆盘的胶合联盟映射到平面上。

## 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。