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

# 数学代写|拓扑学代写TOPOLOGY代考|A Detour

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|A Detour

Sometimes the greatest ideas in mathematics stem from simple, when not trivial, observations. Consider the singleton ${k}$ as topological space. Any space $Y$ is homeomorphic in a natural way to $C({}, Y)$, the space of continuous maps from ${}$ to $Y$ endowed with the compact-open topology. Therefore
$$\pi_0(Y)=\pi_0(C({}, Y))$$ Who says we have to stop at ${}$ ? We may as well fix a locally compact Hausdorff space $X$, consider continuous maps $C(X, Y)$ with the compact-open topology and define the set
$$[X, Y]=\pi_0(C(X, Y))$$
Each homeomorphism $Y \cong Z$ induces a homeomorphism $C(X, Y) \cong C(X, Z)$, hence a bijection $[X, Y] \cong[X, Z]$. So if we wanted to prove that the sphere isn’t homeomorphic to the torus, we could show that $\left[S^2, S^1 \times S^1\right]$ contains one point only, whereas $\left[S^2, S^2\right]$ is a countably infinite space. The idea is certainly fascinating, and Theorem 8.20 implies that two continuous maps $f_0, f_1: X \rightarrow Y$ belong in the same path component in $C(X, Y)$ if and only if they are homotopic.

For technical reasons to be clarified later, it’s more convenient to work with pointed topological spaces, i.e. pairs $\left(X, x_0\right)$ where $x_0 \in X$. We define $\left[\left(X, x_0\right),\left(Y, y_0\right)\right]=$ $\pi_0\left(C\left(\left(X, x_0\right),\left(Y, y_0\right)\right)\right)$, where
$$C\left(\left(X, x_0\right),\left(Y, y_0\right)\right)=\left{f \in C(X, Y) \mid f\left(x_0\right)=y_0\right} .$$

## 数学代写|拓扑学代写TOPOLOGY代考|Path Homotopy

We have already introduced the space of paths in $X$ between points $a, b \in X$
$$\Omega(X, a, b)={\alpha: I \rightarrow X \mid \alpha \text { continuous, } \alpha(0)=a, \alpha(1)=b}$$
We also have defined the product and the inversion
$$\begin{gathered} *: \Omega(X, a, b) \times \Omega(X, b, c) \rightarrow \Omega(X, a, c), \quad \alpha * \beta(t)= \begin{cases}\alpha(2 t) & \text { if } 0 \leq t \leq \frac{1}{2}, \ \beta(2 t-1) & \text { if } \frac{1}{2} \leq t \leq 1 .\end{cases} \ i: \Omega(X, a, b) \rightarrow \Omega(X, b, a), \quad i(\alpha)(t)=\alpha(1-t) . \end{gathered}$$
Note that $i(i(\alpha))=\alpha$ and $i(\alpha * \beta)=i(\beta) * i(\alpha)$.
Definition 11.1 Two paths $\alpha, \beta \in \Omega(X, a, b)$ are path homotopic if there is a continuous map $F: I \times I \rightarrow X$ such that:

$F(t, 0)=\alpha(t), F(t, 1)=\beta(t)$ for every $t \in I$;

$F(0, s)=a, F(1, s)=b$ for every $s \in I$.
Such an $F$ is called a path homotopy between $\alpha$ and $\beta$.

We remark that the notion of path homotopy is more restrictive than that of homotopy of continuous maps. Here we additionally demand that intermediate paths
$$F_s: I \rightarrow X, \quad F_s(t)=F(t, s),$$
have the same initial and end points, for every $s \in I$. We’ll write $\alpha \sim \beta$ to mean that $\alpha$ and $\beta$ are path homotopic (Fig. 11.1).

We know from earlier that homotopic maps define an equivalence relation; the same proof, with minimal changes, also shows that path homotopy is an equivalence relation.

## 数学代写|拓扑学代写TOPOLOGY代考|A Detour

$$\pi_0(Y)=\pi_0(C({}, Y))$$谁说我们必须停在${}$ ?我们也可以固定一个局部紧化的Hausdorff空间$X$，考虑具有紧开拓扑的连续映射$C(X, Y)$并定义集合
$$[X, Y]=\pi_0(C(X, Y))$$

## 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。