## 数学代写|拓扑学代写Topology代考|MATH3061

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Functions on Smooth Manifolds

In previous sections, we introduced topological spaces, including the special case of (smooth) manifolds. Very often, a space can be equipped with continuous functions defined on it. In this section, we focus on real-valued functions of the form $f: X \rightarrow \mathbb{R}$ defined on a topological space $X$, also called scalar functions; see Figure 1.8(a) for the graph of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. Scalar functions appear commonly in practice that describe space/data of interest (e.g., the elevation function defined on the surface of the Earth). We are interested in the topological structures behind scalar functions. In this section, we limit our discussion to nicely behaved scalar functions (called Morse functions) defined on smooth manifolds. Their topological structures are characterized by the so-called critical points which we will introduce below. Later in the book we will also discuss scalar functions on simplicial complex domains, as well as more complex maps defined on a space $X$, for example, a multivariate function $f: X \rightarrow \mathbb{R}^d$

In what follows, for simplicity of presentation, we assume that we consider smooth ( $C^{\infty}$-continuous) functions and smooth manifolds embedded in $\mathbb{R}^d$, even though often we only require the functions (resp. manifolds) to be $C^2$ continuous (resp. $C^2$-smooth).

To provide intuition, let us start with a smooth scalar function defined on the real line, $f: \mathbb{R} \rightarrow \mathbb{R}$; the graph of such a function is shown in Figure 1.8(b). Recall that the derivative of a function at a point $x \in \mathbb{R}$ is defined as
$$D f(x)=\frac{d}{d x} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t)-f(x)}{t} .$$ The value $D f(x)$ gives the rate of change of the value of $f$ at $x$. This can be visualized as the slope of the tangent line of the graph of $f$ at $(x, f(x))$. The critical points of $f$ are the set of points $x$ such that $D f(x)=0$. For a function defined on the real line, there are two types of critical points in the generic case: maxima and minima, as marked in Figure 1.8(b).

## 数学代写|拓扑学代写Topology代考|Morse Functions and Morse Lemma

From the first-order derivatives of a function we can determine critical points. We can learn more about the “type” of the critical points by inspecting the second-order derivatives of $f$.

A critical point $x$ of $f$ is nondegenerate if its Hessian matrix, Hessian $(x)$, is nonsingular (has nonzero determinant); otherwise, it is a degenerate critical point.

For example, consider $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^3-3 x y^2$. The origin $(0,0)$ is a degenerate critical point often referred to as a “monkey saddle:” see Figure 1.9(d), where the graph of the function around $(0,0)$ goes up and down three times (instead of twice as for a nondegenerate saddle shown in Figure 1.9b). It turns out that, as a consequence of the Morse Lemma below, nondegenerate critical points are always isolated whereas the degenerate ones may not be so. A simple example is $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^2$, where all points on the $y$-axis are degenerate critical points. The local neighborhood of nondegenerate critical points can be completely characterized by the following Morse Lemma.

Proposition 1.2. (Morse Lemma) Given a smooth function $f: M \rightarrow \mathbb{R}$ defined on a smooth $m$-manifold $M$, let $p$ be a nondegenerate critical point of $f$. Then there is a local coordinate system in a neighborhood $U(p)$ of $p$ so that (i) the coordinate of $p$ is $(0,0, \ldots, 0)$, and (ii) locally for every point $x=\left(x_1, x_2, \ldots, x_m\right)$ in neighborhood $U(p)$,
$f(x)=f(p)-x_1^2-\cdots-x_s^2+x_{s+1}^2 \cdots+x_m^2, \quad$ for some $s \in[0, m]$.
The number s of minus signs in the above quadratic representation of $f(x)$ is called the index of the critical point $p$.

A critical point $x$ of $f$ is nondegenerate if its Hessian matrix, Hessian $(x)$, is nonsingular (has nonzero determinant); otherwise, it is a degenerate critical point.

For example, consider $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^3-3 x y^2$. The origin $(0,0)$ is a degenerate critical point often referred to as a “monkey saddle:” see Figure 1.9(d), where the graph of the function around $(0,0)$ goes up and down three times (instead of twice as for a nondegenerate saddle shown in Figure 1.9b). It turns out that, as a consequence of the Morse Lemma below, nondegenerate critical points are always isolated whereas the degenerate ones may not be so. A simple example is $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^2$, where all points on the $y$-axis are degenerate critical points. The local neighborhood of nondegenerate critical points can be completely characterized by the following Morse Lemma.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Functions on Smooth Manifolds

$$D f(x)=\frac{d}{d x} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t)-f(x)}{t} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MTH3130

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Maps, Homeomorphisms, and Homotopies

The equivalence of two topological spaces is determined by how the points that comprise them are connected. For example, the surface of a cube can be deformed into a sphere without cutting or gluing it because they are connected the same way. They have the same topology. This notion of topological equivalence can be formalized via functions that send the points of one space to points of the other while preserving the connectivity.

This preservation of connectivity is achieved by preserving the open sets. A function from one space to another that preserves the open sets is called a continuous function or a map. Continuity is a vehicle to define topological equivalence, because a continuous function can send many points to a single point in the target space, or send no points to a given point in the target space. If the former does not happen, that is, when the function is injective, we call it an embedding of the domain into the target space. True equivalence is given by a homeomorphism, a bijective function from one space to another which has continuity as well as a continuous inverse. This ensures that open sets are preserved in both directions.

Definition 1.15. (Continuous function; Map) A function $f: \mathbb{T} \rightarrow \mathbb{U}$ from the topological space $\mathbb{T}$ to another topological space $\mathbb{U}$ is continuous if for every open set $Q \subseteq \mathbb{U}, f^{-1}(Q)$ is open. Continuous functions are also called maps.
Definition 1.16. (Embedding) A map $g: \mathbb{T} \rightarrow \mathbb{U}$ is an embedding of $\mathbb{V}$ into $\mathbb{U}$ if $g$ is injective.

A topological space can be embedded into a Euclidean space by assigning coordinates to its points so that the assignment is continuous and injective. For example, drawing a triangle on paper is an embedding of $\mathbb{S}^1$ into $\mathbb{R}^2$. There are topological spaces that cannot be embedded into a Euclidean space, or even into a metric space – these spaces cannot be represented by any metric.

Next we define a homeomorphism that connects two spaces that have essentially the same topology.

Definition 1.17. (Homeomorphism) Let $\mathbb{T}$ and $\mathbb{U}$ be topological spaces. A homeomorphism is a bijective map $h: \mathbb{T} \rightarrow \mathbb{U}$ whose inverse is continuous too.

Two topological spaces are homeomorphic if there exists a homeomorphism between them.

## 数学代写|拓扑学代写Topology代考|Manifolds

A manifold is a topological space that is locally connected in a particular way. A 1-manifold has this local connectivity looking like a segment. A 2manifold (with boundary) has the local connectivity looking like a complete or partial disk. In layman’s terms, a 2-manifold has the structure of a piece of paper or rubber sheet, possibly with the houndaries glued together to form a closed surface – a category that includes disks, spheres, tori, and Möbius bands.

Definition 1.22. (Manifold) A topological space $M$ is an m-manifold, or simply a manifold, if every point $x \in M$ has a neighborhood homeomorphic to $\mathbb{B}_o^m$ or $\mathbb{H}^m$. The dimension of $M$ is $m$.

Every manifold can be partitioned into boundary and interior points. Observe that these words mean very different things for a manifold than they do for a metric space or topological space.

Definition 1.23. (Boundary; Interior) The interior Int $M$ of an $m$-manifold $M$ is the set of points in $M$ that have a neighborhood homeomorphic to $\mathbb{B}_o^m$. The boundary $\mathrm{Bd} M$ of $M$ is the set of points $M \backslash \operatorname{Int} M$. The boundary $\operatorname{Bd} M$, if not empty, consists of the points that have a neighborhood homeomorphic to $\mathbb{H}^m$. If $\mathrm{Bd} M$ is the empty set, we say that $M$ is without boundary.

A single point, a 0 -ball, is a 0 -manifold without boundary according to this definition. The closed disk $\mathbb{B}^2$ is a 2-manifold whose interior is the open disk $\mathbb{B}_o^2$ and whose boundary is the circle $\mathbb{S}^1$. The open disk $\mathbb{B}_o^2$ is a 2-manifold whose interior is $\mathbb{B}_o^2$ and whose boundary is the empty set. This highlights an important difference between Definitions $1.13$ and $1.23$ of “boundary”: when $\mathbb{B}_o^2$ is viewed as a point set in the space $\mathbb{R}^2$, its boundary is $\mathbb{S}^1$ according to Definition 1.13; but viewed as a manifold, its boundary is empty according to Definition 1.23. The boundary of a manifold is always included in the manifold.

The open disk $\mathbb{B}_o^2$, the Euclidean space $\mathbb{R}^2$, the sphere $\mathbb{S}^2$, and the torus are all connected 2-manifolds without boundary. The first two are homeomorphic to each other, but the last two are not. The sphere and the torus in $\mathbb{R}^3$ are compact (bounded and closed with respect to $\mathbb{R}^3$ ) whereas $\mathbb{B}_o^2$ and $\mathbb{R}^2$ are not.

A $d$-manifold, $d \geq 2$, can have orientations whose formal definition we skip here. Informally, we say that a 2-manifold $M$ is non-orientable if, starting from a point $p$, one can walk on one side of $M$ and end up on the opposite side of $M$ upon returning to $p$. Otherwise, $M$ is orientable. Spheres and balls are orientable, whereas the Möbius band in Figure 1.7(a) is a non-orientable 2-manifold with boundary.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Manifolds

1.13；但作为流形来看，根据定义 1.23，它的边界是空的。流形的边界总是包含在流形中。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH3402

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Topological Space

The basic object in a topological space is a ground set whose elements are called points. A topology on these points specifies how they are connected by listing what points constitute a neighborhood – the so-called open set.

The expression “rubber-sheet topology” commonly associated with the term “topology” exemplifies this idea of connectivity of neighborhoods. If we bend and stretch a sheet of rubber, it changes shape but always preserves the neighborhoods in terms of the points and how they are connected.

We first introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas – manifolds, homeomorphism, isotopy, and other maps – used later to study algorithms for topological data analysis. Homeomorphisms, for example, offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed intoo aa shape without ever colliding with itself.

Perhaps it is more intuitive to understand the concept of topology in the presence of a metric because then we can use the metric balls such as Euclidean balls in a Euclidean space to define neighborhoods – the open sets. Topological spaces provide a way to abstract out this idea without a metric or point coordinates, so they are more general than metric spaces. In place of a metric, we encode the connectivity of a point set by supplying a list of all of the open sets. This list is called a system of subsets of the point set. The point set and its system together describe a topological space.

## 数学代写|拓扑学代写Topology代考|Metric Space Topology

Metric spaces are a special type of topological space commonly encountered in practice. Such a space admits a metric that specifies the scalar distance between every pair of points satisfying certain axioms.

Definition 1.8. (Metric space) A metric space is a pair ( $\mathbb{T}, d)$ where $\mathbb{T}$ is a set and $d$ is a distance function $d: \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ satisfying the following properties:

• $\mathrm{d}(p, q)=0$ if and only if $p=q$ for all $p \in \mathbb{T}$;
• $\mathrm{d}(p, q)=\mathrm{d}(q, p)$ for all $p, q \in \mathbb{\pi}$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ for all $p, q, r \in \mathbb{T}$.
It can be shown that the three axioms above imply that $\mathrm{d}(p, q) \geq 0$ for every pair $p, q \in \mathbb{T}$. In a metric space $\mathbb{T}$, an open metric ball with center $c$ and radius $r$ is defined to be the point set $B_o(c, r)={p \in \mathbb{T}: \mathrm{d}(p, c)<r}$. Metric balls define a topology on a metric space.

Definition 1.9. (Metric space topology) Given a metric space $\mathbb{T}$, all metric balls $\left{B_o(c, r) \mid c \in \mathbb{T}\right.$ and $\left.0<r \leq \infty\right}$ and their union constituting the open sets define a topology on $\mathbb{T}$.

All definitions for general topological spaces apply to metric spaces with the above defined topology. However, we give alternative definitions using the concept of limit points which may be more intuitive.

As we have mentioned already, the heart of topology is the question of what it means for a set of points to be connected. After all, two distinct points cannot be adjacent to each other; they can only be connected to one another by passing through uncountably many intermediate points. The idea of limit points helps express this concept more concretely, specifically in the case of metric spaces. We use the notation $\mathrm{d}(\cdot, \cdot)$ to express minimum distances between point sets $P, Q \subseteq \mathbb{T}:$
\begin{aligned} & \mathrm{d}(p, Q)=\inf {\mathrm{d}(p, q): q \in Q} \ & \mathrm{d}(P, Q)=\inf {\mathrm{d}(p, q): p \in P, q \in Q} \end{aligned}

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Metric Space Topology

• $\mathrm{d}(p, q)=0$ 当且仅当 $p=q$ 对所有人 $p \in \mathbb{T}$;
• $\mathrm{d}(p, q)=\mathrm{d}(q, p)$ 对所有人 $p, q \in \pi$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ 对所有人 $p, q, r \in \mathbb{T}$.
可以证明，上面的三个公理意味着 $\mathrm{d}(p, q) \geq 0$ 每对 $p, q \in \mathbb{T}$. 在度量空间 $\mathbb{T}$, 一个中心为空心的公 制球 $c$ 和半径 $r$ 被定义为点集 $B_o(c, r)=p \in \mathbb{T}: \mathrm{d}(p, c)<r$. 度量球定义度量空间上的拓扑。
定义 1.9。 (度量空间拓扑) 给定一个度量空间 $\mathbb{T}$ ，所有公制球 个拓扑T⿺丄⺊.
一般拓扑空间的所有定义都适用于具有上述定义拓扑的度量空间。但是，我们使用可能更直观的极限点 概念给出替代定义。
正如我们已经提到的，拓扑的核心问题是连接一组点意味着什么。毕竟，两个不同的点不能彼此相邻； 它们只能通过无数个中间点才能相互连接。极限点的概念有助于更具体地表达这个概念，特别是在度量 空间的情况下。我们使用符号 $\mathrm{d}(\cdot, \cdot)$ 表达点集之间的最小距离 $P, Q \subseteq \mathbb{T}$ :
$$\mathrm{d}(p, Q)=\inf \mathrm{d}(p, q): q \in Q \quad \mathrm{~d}(P, Q)=\inf \mathrm{d}(p, q): p \in P, q \in Q$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH3531

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Continuous Maps and Homeomorphisms

The concept of continuous maps often appears as the first notion in a book on calculus or analysis. The epsilon-delta definition of continuity of a function on the real line was first given by Bolzano in 1817 . Generalizing it without using the metric structure, one may define continuous maps between topological spaces. In Section 3.1, we shall formulate a definition of continuity that includes all kinds of continuities as special cases and study various properties of this general continuity. The essential concept of homeomorphisms between topological spaces comes out in Section $3.2$.
In this book, the words map and function are used interchangeably.
Let $X$ be a topological space with a subset $A \subseteq X$. The identity map is the map
\text { id: } \begin{aligned} X & \rightarrow X \ x & \mapsto x . \end{aligned}
It is a notion on the level of sets, rather than the level of topological spaces. As a result, the identity map from a topological space $A=(X, \Omega)$ to another space $B=\left(X, \Omega_{B}\right)$ is not continuous when $B$ has an open set that is not open in $A$. The inclusion from $A$ to $X$ is the map
\begin{aligned} \iota: A & \rightarrow X \ x & \mapsto x . \end{aligned}
Let $Y$ be a topological space with a subset $B \subseteq Y$. A map
$$f: X \rightarrow Y$$ is a constant map if the image $f(X)$ is a singleton. Let
$$f(A) \subseteq B .$$

## 数学代写|拓扑学代写Topology代考|Continuous maps

Definition 3.1. A map
$$f:\left(X, \Omega_{X}\right) \rightarrow\left(Y, \Omega_{Y}\right)$$
between topological spaces is open if every open set of $X$ is mapped to an open set of $Y$. The map $f$ is closed if every closed set of $X$ is mapped to a closed set of $Y$. The map $f$ is continuous if the preimage of any open set of $Y$ is an open set of $X$. It is continuous at a point $x \in X$ if the set $f^{-1}(V)$ is a neighborhood of $x$ for every neighborhood $V$ of $f(x)$.

Example $3.2$ exhibits that the notion of continuity in topology and that in calculus are different.
Example 3.2. Consider topological spaces $\left(X, \Omega_{\mathbb{R}}\right)$ and $\left(Y, \Omega_{Y}\right)$, where
$$X=Y=[0,2] .$$
If $\Omega_{Y}=\Omega_{\mathbb{R}}$, then the function
\begin{aligned} f: X & \rightarrow Y \ x & \mapsto \begin{cases}x, & \text { if } x \in[0,1), \ 3-x, & \text { if } x \in[1,2]\end{cases} \end{aligned}
is not continuous. This coincides with what we learnt from analysis. If the topology $\Omega_{Y}$ is induced from the arrow, then the function
\begin{aligned} f: X & \rightarrow Y \ x & \mapsto \begin{cases}x, & \text { if } x \in[0,1], \ x+1, & \text { if } x \in(1,2]\end{cases} \end{aligned}
is continuous! This is different from what we know from analysis.

## 数学代写|拓扑学代写Topology代考|Continuous Maps and Homeomorphisms

id: $X \rightarrow X x \quad \mapsto x .$

$$\iota: A \rightarrow X x \quad \mapsto x .$$

$$f: X \rightarrow Y$$

$$f(A) \subseteq B$$

## 数学代写|拓扑学代写Topology代考|Continuous maps

$$f:\left(X, \Omega_{X}\right) \rightarrow\left(Y, \Omega_{Y}\right)$$

$$X=Y=[0,2] .$$

$$f: X \rightarrow Y x \mapsto{x, \quad \text { if } x \in[0,1), 3-x, \quad \text { if } x \in[1,2]$$

$$f: X \rightarrow Y x \quad \mapsto{x, \quad \text { if } x \in[0,1], x+1, \quad \text { if } x \in(1,2]$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH4204

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Exercises 2

2.1) The set $K \cup{0}$ is closed in the real line, where $K$ is the set of unit fractions.
2.2) A topology is defined by assigning the open sets of a set $X$ in Definition 2.1. Can it be defined by assigning the closed sets of $X$ ?
2.3) Can we define a topology by assigning the neighborhoods?
2.4) Find a smallest topological space which is neither discrete nor indiscrete. Is it unique?
2.5) Is there a topology which is both a particular point topology and an excluded point topology?
2.6) Find a topological space $(X, \Omega)$ with a set $A \subset X$ satisfying
(a) $A$ is neither open nor closed;
(b) $A$ is the union of an infinite number of closed sets; and
(c) $A$ is the intersection of an infinite number of open sets.
2.7) Dèscribè all néighborhoods of a point in
(a) a discrete space;
(b) an indiscrete space;
(c) the arrow;
(d) a particular point space;
(e) the Sierpiński space.
2.8) The terminology “basis” in linear algebra and the terminology “base” in topology, as English words, have the same plural form “bases”. What can we say about their differences as mathematical concepts?

## 数学代写|拓扑学代写Topology代考|Selected Solutions

2.1) The complement of the set is the union
$$(-\infty, 0) \cup\left{\left(\frac{1}{n+1}, \frac{1}{n}\right): n \in \mathbb{Z}^{+}\right} \cup(1,+\infty)$$
of open sets.
2.2) Yes. Here is a list of axioms for assigning the closed sets:
(a) the empty set $\emptyset$ and $X$ are closed;
(b) the union of any finite number of closed sets is closed;
(c) the intersection of any collection of closed sets is closed.
2.3) Yes. Here is a list of axioms for assigning the collection $\mathcal{N}{x}$ of neighborhoods to each point $x$ : (a) If $x \in X$ and $U \in \mathcal{N}{x}$, then $x \in U$.
(b) If $x \in X$ and $U_{1}, U_{2} \in \mathcal{N}{x}$, then $$U{1} \cap U_{2} \in \mathcal{N}{x}$$ (c) If $x \in X, U \in \mathcal{N}{x}$, and $U \subset V$, then
$$V \in \mathcal{N}{x} .$$ (d) If $x \in X$ and $U \in \mathcal{N}{x}$, then
$$\left{y \in V: V \in \mathcal{N}{y}\right} \in \mathcal{N}{x} .$$
2.6) The interval $[0,1)$ in $\mathbb{R}$.
2.8) In linear algebra, a set $B$ of elements in a vector space $V$ is called a basis if every element of $V$ may be written in a unique way as a finite linear combination of elements of $B$. In contrast to a basis, it is not necessary for a base to be maximal. For example, any open set can be added to a base to form a new base. Moreover, a topological space may have disjoint bases of distinct sizes. For example, the standard topology on the real line has a base of all open intervals with rational ends, and another base of all open intervals with irrational ends.

## 数学代写|拓扑学代写Topology代考|Exercises 2

2.1) 集合 $K \cup 0$ 在实线中闭合，其中 $K$ 是单位分数的集合。
2.2）拓扑是通过分配一个集合的开集来定义的 $X$ 在定义 $2.1$ 中。是否可以通过分配闭集来定义 $X$ ?
2.3) 我们可以通过分配邻域来定义拓扑吗?
2.4) 找到一个既不是离散也不是不离散的最小拓扑空间。它是独一无二的吗?
2.5) 是否存在既是特定点拓扑又是排除点拓扑的拓扑?
2.6) 寻找拓扑空间 $(X, \Omega)$ 用一套 $A \subset X$ 满足
（一） $A$ 既不开放也不封闭；
(b) $A$ 是无限多闭集的并集；(
c) $A$ 是无限个开集的交集。
2.7) 描述
(a) 离散空间中一个点的所有邻域；
(b) 杂乱无章的空间；
(c) 箭头；
(d) 一个特定的点空间；
(e) 谢尔宾斯基空间。
2.8) 线性代数中的术语“基础”和拓扑学中的术语“基础”，作为英文单词，具有相同的复数形式“基础”。对于它们作为 数学概念的差异，我们能说些什么?

## 数学代写|拓扑学代写Topology代考|Selected Solutions

2.1) 集合的补集是并集
$(-$ linfty, 0) \cup\left } { \backslash \text { left(\frac } { 1 } { n + 1 } , \backslash \text { frac } { 1 } n } \backslash \text { right): } n \backslash \text { in } \backslash \text { mathbb } { Z } ^ { \wedge } { + } \backslash \text { right } } \backslash c u p ( 1 , + \backslash i n f t y )

2.2) 是的。以下是分配闭集的公理列表:
(a) 空集 $\emptyset$ 和 $X$ 已关闭;
(b) 任何有限数量的封闭集的并集是封闭的；
(c) 任何闭集的交集都是闭集。
2.3) 是的。这是分配集合的公理列表 $\mathcal{N} x$ 每个点的邻域 $x$ : (a) 如果 $x \in X$ 和 $U \in \mathcal{N} x$ ，然后 $x \in U$.
(b) 如果 $x \in X$ 和 $U_{1}, U_{2} \in \mathcal{N} x$ ，然后
$$U 1 \cap U_{2} \in \mathcal{N} x$$
(c) 如果 $x \in X, U \in \mathcal{N} x$ ，和 $U \subset V$ ，然后
$$V \in \mathcal{N} x .$$
(d) 如果 $x \in X$ 和 $U \in \mathcal{N} x$ ，然后
\left } { y \backslash \text { in } \mathrm { V } : \mathrm { V } \backslash \text { in } \backslash \text { mathcal } { \mathrm { N } } { \mathrm { y } } \backslash \text { right } } \backslash \text { in } \backslash \text { mathcal } { \mathrm { N } } { \mathrm { x } } \text { 。 }
2.6) 区间 $[0,1)$ 在 $\mathbb{R}$.
2.8) 在线性代数中，一个集合 $B$ 向量空间中的元素 $V$ 如果每个元素都称为基 $V$ 可以用一种独特的方式写成元素的有 限线性组合 $B$. 与基相比，基不一定是最大的。例如，可以将任何开集添加到碱基以形成新碱基。此外，拓扑空间可 能具有不同大小的不相交基。例如，实线上的标准拓扑有一个基是所有开区间的有理端点，另一个基是所有开区间 的无理端点。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH6204

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Metrics and the metric topology

This section is devoted to a special kind of topological spaces – the metric spaces. A parallel context is Sutherland’s book [38], which puts an emphasis on metric spaces.
The Cartesian product of two sets $X$ and $Y$ is the set
$$X \times Y={(x, y): x \in X, y \in Y} .$$
It is also called the direct product. The Cartesian product of $n$ copies of a set $\bar{X}$ is usually denoted as $\bar{X}^{n}$. A function
$$f: X^{n} \rightarrow \mathbb{C}$$
is said to be symmetric if
$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=f\left(x_{i_{1}}, x_{i_{2}}, \ldots, x_{i_{n}}\right)$$
for any rearrangement $\left(i_{1}, i_{2}, \ldots, i_{n}\right)$ of the subscripts $1,2, \ldots, n$, where $\mathbb{C}$ is the set of complex numbers.

## 数学代写|拓扑学代写Topology代考|Topological Spaces

Exercises $2.19$ and $2.20$ give other two ways of self-production of metrics.
The concept of norm in linear algebra, functional analysis and related areas has a close relationship with metrics. Let $X$ be a set. A function
$$f: X \rightarrow \mathbb{R}$$
$$f(x+y) \leq f(x)+f(y)$$
for any $x, y \in X$. Let $V$ be a vector space over $\mathbb{R}$. A seminorm on $V$ is a function
$$|\cdot|: V \rightarrow \mathbb{R}$$
that is subadditive and absolutely scalable:
$$|\lambda v|=|\lambda| \cdot|v|, \quad \forall \lambda \in \mathbb{R}, \forall v \in V .$$
The trivial seminorm is the function that maps every vector to zero. Any seminorm is positive semi-definite, i.e.,
$$|v| \geq 0, \quad \forall v \in V .$$
In fact, the absolute scalability implies
$$|0|=0 \quad \text { and } \quad|-v|=|v| .$$
Taking $u=-v$ in the subadditivity condition that
$$|u+v| \leq|u|+|v|$$
we obtain $|v| \geq 0$. A seminorm is a norm if it is definite, that is, if it satisfies the implication
$$|v|=0 \Rightarrow v=0$$

## 数学代写|拓扑学代写Topology代考|Metrics and the metric topology

$$X \times Y=(x, y): x \in X, y \in Y .$$

$$f: X^{n} \rightarrow \mathbb{C}$$

$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=f\left(x_{i_{1}}, x_{i_{2}}, \ldots, x_{i_{n}}\right)$$

## 数学代写|拓扑学代写Topology代考|Topological Spaces

$$f: X \rightarrow \mathbb{R}$$

$$f(x+y) \leq f(x)+f(y)$$

$$|\cdot|: V \rightarrow \mathbb{R}$$

$$|\lambda v|=|\lambda| \cdot|v|, \quad \forall \lambda \in \mathbb{R}, \forall v \in V .$$

$$|v| \geq 0, \quad \forall v \in V .$$

$$|0|=0 \quad \text { and } \quad|-v|=|v| .$$

$$|u+v| \leq|u|+|v|$$

$$|v|=0 \Rightarrow v=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH3402

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Topological structures

It took a long time in forming the following definition of a topological space. According to Munkres [30, page 75], “mathematicians … wanted, of course, a definition that was as broad as possible, so that it would include as special cases all the various examples that were useful in mathematics … but they also wanted the definition to be narrow enough that the standard theorems about these familiar spaces would hold for topological spaces in general.”
Definition 2.1. A topology on a set $X$ is a collection $\Omega \in 2^{X}$ satisfying the axioms
(1) $\emptyset \in \Omega$ and $X \in \Omega$;
(2) the union of any members of $\Omega$ lies in $\Omega$; and
(3) the intersection of any two members of $\Omega$ lies in $\Omega$.
A topological space is a pair $(X, \Omega)$, where $\Omega$ is a topology on $X$. An open set in $(X, \Omega)$ is a member in $\Omega$. A closed set in $(X, \Omega)$ is a subset $A \subseteq X$ such that $X \backslash A \in \Omega$. A clopen set in $(X, \Omega)$ is a subset $A \subseteq X$ that is both closed and open. A neighborhood of a point $p \in X$ is a subset $U \subseteq X$ such that $p \in O \subseteq U$ for some open set $O \in \Omega$.

Throughout this book, we use the letter $\Omega$ to denote an open set, ${ }^{4}$ and the letter $U$ a neighborhood since it is the first letter of the German word “umgebung”, which means neighborhood. The definition of a neighborhood follows the Nicolas Bourbaki group $5[11,12]$.
Example 2.2. Are the following pairs $(X, \Omega)$ topological spaces?
(1) $X=\mathbb{R}$ is the set of real numbers, and
$$\Omega={\text { infinite subsets of } \mathbb{R}} \cup{\emptyset} .$$
(2) $X=\mathbb{R}^{2}$ is the set of points on the plane, and $\Omega={$ opén disks céntérẽd át thé úigin $} \cup{\emptyset, X}$.

## 数学代写|拓扑学代写Topology代考|Point position with respect to a set

We introduce some terminologies on point positions with respect to a set.
Definition 2.14. Let $X$ be a topological space with a subset $A \subseteq X$. $\mathrm{A}$ limit point of $A$ is a point $p \in X$ such that
$$(A \backslash{P}) \cap U \neq \emptyset$$
for any neighborhood $U$ of $p$. An isolated point of $A$ is a point $p \in A$ such that
$(A \backslash{p}) \cap U=\emptyset$
for some neighborhood $U$ of $p$. The set $A$ is perfect if it is closed and has no isolated points. The closure of $A$ is the union of $A$ and its limit points, denoted $\bar{A}$, i.e.,
$$\bar{A}={p \in X: N \cap A \neq \emptyset \text { for any neighborhood } N \text { of } p} .$$
An adherent point of $A$ is a point in the closure $\bar{A}$. An interior point of $A$ is a point having a neighborhood in $A$. The interior of $A$, denoted $A^{\circ}$, is the set of interior points. An exterior point of $A$ is a point that has a neighborhood in the complement
$$A^{c}=X \backslash A .$$
The exterior of $A$ is the set $\left(A^{c}\right)^{\circ}$ of exterior points. A boundary point of $A$ is a point such that each of its neighborhood meets both $A$ and $A^{c}$. The boundary of $A$ is the set $\bar{A} \backslash A^{\circ}$ of boundary points, denoted $\partial A$. When one expresses the closure of $A$ with an emphasis on the operation of taking closure, he may use the symbol $\mathrm{Cl}(A)$ (or $\mathrm{Cl}_{X} A$ when there is a possibility of confusion). The symbols $\operatorname{Int}(A), \operatorname{Ext}(A)$, and $\operatorname{Bd}(A)$ are used in the similar manner.

## 数学代写|拓扑学代写Topology代考|Topological structures

(1) $\emptyset \in \Omega$ 和 $X \in \Omega$;
(2) 任何成员的工会 $\Omega$ 在于 $\Omega$; (
3) 任意两个成员的交集 $\Omega$ 在于 $\Omega$.

(1) $X=\mathbb{R}$ 是实数的集合，并且
$$\Omega=\text { infinite subsets of } \mathbb{R} \cup \emptyset .$$

(2) $X=\mathbb{R}^{2}$ 是平面上的点集，并且 $\Omega=$ \$opéndiskscéntérẽdatthéúigin$\$\cup \emptyset, X$.

## 数学代写|拓扑学代写Topology代考|Point position with respect to a set

$$(A \backslash P) \cap U \neq \emptyset$$

$(A \backslash p) \cap U=\emptyset$

$$\bar{A}=p \in X: N \cap A \neq \emptyset \text { for any neighborhood } N \text { of } p .$$

$$A^{c}=X \backslash A \text {. }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH6204

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Electromagnetic fields

In the $1860 \mathrm{~s}$, the Scottish physicist James Clerk Maxwell gathered together all that was known at the time about electricity and magnetism and showed that it all followed from a small set of equations now known as the Maxwell equations. In modern vector notation and SI units, the differential form of these laws is given by
$$\begin{gathered} \nabla \cdot \boldsymbol{E}=\frac{\rho}{\epsilon_{0}}, \quad \nabla \cdot \boldsymbol{B}=0 \ \nabla \times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}, \quad \nabla \times \boldsymbol{B}=\mu_{0}\left(J+\epsilon_{0} \frac{\partial \boldsymbol{E}}{\partial t}\right) \end{gathered}$$
The electric and magnetic fields can be computed from the scalar potential $\phi(r, t)$ and the vector potential $A(\boldsymbol{r}, t)$,
$$\boldsymbol{B}=\nabla \times \boldsymbol{A} \quad \text { and } \quad \boldsymbol{E}=-\nabla \phi-\frac{\partial \boldsymbol{A}}{d t} .$$
The potentials $\phi$ and $\boldsymbol{A}$ are not entirely well-defined: for any function $f(\boldsymbol{r}, t)$, the gauge transformations
$$A \rightarrow A+\nabla f \quad \text { and } \quad \phi \rightarrow \phi-\frac{\partial f}{\partial t}$$
leave $\boldsymbol{E}$ and $\boldsymbol{B}$ unchanged, along with all other physically measurable quantities. This ambiguity in the potentials is sometimes useful, since it can often be utilized to put them into a form that simplifies a given problem. However, it also introduces conceptual difficulties and raises the question of whether the potentials are physically ‘real’ in the same way the directly measurable $\boldsymbol{E}$ and $\boldsymbol{B}$ fields are. We will see later that the gauge invariance in fact has geometric and topological meaning.

## 数学代写|拓扑学代写Topology代考|Electromagnetic potentials and gauge invariance

Returning to the gauge potential $A_{\mu}$ defined above, the effect of the electromagnetic field acting on a particle of charge $q$ may be introduced via the minimal coupling principle, replacing the free-particle four momentum $p_{\mu}=-i \hbar \partial / \partial x_{\mu}$ by the canonical momentum
$$p_{\mu}=-i \hbar\left(\partial / \partial x_{\mu}\right)+q A_{\mu}$$
Here we are using relativistic four-vector notation, where $\mu=0$ corresponds to the time-like component and $\mu=1,2,3$ are the space-like components:
$$\begin{gathered} A_{\mu}={\phi, \boldsymbol{A}} \ \partial_{\mu}=\frac{\partial}{\partial x_{\mu}}=\left{\frac{\partial}{\partial t}, \nabla\right} \ p_{\mu}={E, \boldsymbol{p}} . \end{gathered}$$

The Schrödinger equation is then of the form
$$\left(\frac{1}{2 m}(i \hbar \nabla+q A)^{2}+q \phi\right) \psi(\boldsymbol{x}, t)=i \hbar \frac{\partial}{\partial t} \psi(x, t)$$
Note that the transition to the canonical momentum may also be viewed as starting from the field-free Schrödinger equation,
$$\frac{\hbar^{2}}{2 m} V^{2} \psi(x, t)=i \hbar \frac{\partial}{\partial t} \psi(x, t),$$
and replacing the ordinary derivatives $\partial_{\mu} \equiv \partial / \partial x_{\mu}(\mu=0,1,2,3)$ by the covariant derivatives
$$D_{\mu}=\partial_{\mu}+\frac{i q}{\hbar} A_{\mu}$$

## 数学代写|拓扑学代写Topology代考|Electromagnetic fields

$$\nabla \cdot \boldsymbol{E}=\frac{\rho}{\epsilon_{0}}, \quad \nabla \cdot \boldsymbol{B}=0 \nabla \times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}, \quad \nabla \times \boldsymbol{B}=\mu_{0}\left(J+\epsilon_{0} \frac{\partial \boldsymbol{E}}{\partial t}\right)$$

$$\boldsymbol{B}=\nabla \times \boldsymbol{A} \quad \text { and } \quad \boldsymbol{E}=-\nabla \phi-\frac{\partial \boldsymbol{A}}{d t} .$$

$$A \rightarrow A+\nabla f \quad \text { and } \quad \phi \rightarrow \phi-\frac{\partial f}{\partial t}$$

## 数学代写|拓扑学代写Topology代考|Electromagnetic potentials and gauge invariance

$$p_{\mu}=-i \hbar\left(\partial / \partial x_{\mu}\right)+q A_{\mu}$$

$$\left(\frac{1}{2 m}(i \hbar \nabla+q A)^{2}+q \phi\right) \psi(\boldsymbol{x}, t)=i \hbar \frac{\partial}{\partial t} \psi(x, t)$$

$$\frac{\hbar^{2}}{2 m} V^{2} \psi(x, t)=i \hbar \frac{\partial}{\partial t} \psi(x, t),$$

$$D_{\mu}=\partial_{\mu}+\frac{i q}{\hbar} A_{\mu}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MATH3402

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Aharanov–Bohm effect

A second demonstration of the importance of topology in physics came with the Aharonov-Bohm effect [8]. Consider a charged particle moving in the vicinity of a current carrying solenoid. There is a magnetic field $\boldsymbol{B} \neq 0$ inside the solenoid, but the field vanishes outside. The vector potential, $\boldsymbol{A}$, however is nonzero everywhere, inside and out. Prior to the rise of the Aharonov-Bohm effect, it was believed that the field $\boldsymbol{B}$ was the physically important variable and that $\boldsymbol{A}$ was simply a mathematical convenience of no physical significance. However, Aharonov and Bohm showed that when the particle circles the solenoid in a closed loop $\mathcal{C}$, staying entirely in the $\boldsymbol{B}=0$ region, there is nevertheless a phase shift given by
$$\Delta \phi=\frac{e}{\hbar c} \int_{\mathcal{C}} \boldsymbol{A}(\boldsymbol{r}) \cdot d \boldsymbol{l}=\frac{e}{\hbar c} \int_{\mathcal{S}} \boldsymbol{B} \cdot d \boldsymbol{s},$$
and that this shift is an integer multiple of $2 \pi$. The existence of the Aharonov-Bohm effect was verified in an experiment by Chambers in 1960 [9].

The solenoid contains a singularity in the vector potential. One can therefore view the solenoid as a hole in the space of allowed field configurations. The quantization arises from the topological fact that curves in $A$-space that enclose the solenoid are non-contractible. The integer $n$ here counts the number of times the loop encloses the singularity: it is a winding number. This winding number characterizes the distinct homotopy classes (see chapters 3 and 5) of the field. The Aharanov-Bohm phase accumulated as the electron circles the solenoid is an example of the geometric Berry phase to be discussed in chapter 9 .

## 数学代写|拓扑学代写Topology代考|Topology in optics

By the 1990s and 2000s, many of the topology-related structures previously found in other areas of physics began to come up in optics. For example, vortices and vortex lines, winding numbers and linking numbers, and even non-orientable Möbius strips have all made appearances in various areas of optics. Further, the Aharonov-Bohm effect is a special case of the geometric or Berry phase; the first known description of a geometric phase appeared in a study of polarization optics in the 1950s, although its significance was not widely recognized for decades.

All of these topics will be described in coming chapters. The range of optical phenomena in which topology plays a role has become large, so in a book of this size some of them will necessarily be treated only in the briefest of terms, but hopefully enough of a flavor will be given to interest the reader in pursuing a deeper study via the provided references.

As general references to the broader background material, we list a few useful texts here. Many excellent introductions to algebraic and differential topology may be found, including [10-15]. Numerous reviews covering applications of topology to gauge field theory, particle physics, and condensed matter physics also exist, which physicists and engineers may find more accessible; these include [16-20]. The history of topology and of its applications in physics are reviewed in [21] and [22], respectively.

## 数学代写|拓扑学代写Topology代考|Aharanov–Bohm effect

Aharonov-Bohm 效应 [8] 再次证明了拓扑在物理学中的重要性。考虑在载流螺线管附近移动的带电粒子。有磁场 $\boldsymbol{B} \neq 0$ 在螺线管内，但场在外面消失。向量势， $\boldsymbol{A}$, 然而，无论从内部还是外部，它都是非零的。在阿哈罗诺夫玻姆效应兴起之前，人们认为场 $\boldsymbol{B}$ 是物理上重要的变量，并且 $\boldsymbol{A}$ 只是一种没有物理意义的数学便利。然而，

Aharonov 和 Bohm 表明，当粒子在闭合回路中环绕螺线管时 $\mathcal{C}{1}$ 完全停留在 $\boldsymbol{B}=0$ 区域，仍然存在由下式给出的 相移 $$\Delta \phi=\frac{e}{\hbar c} \int{\mathcal{C}} \boldsymbol{A}(\boldsymbol{r}) \cdot d \boldsymbol{l}=\frac{e}{\hbar c} \int_{\mathcal{S}} \boldsymbol{B} \cdot d \boldsymbol{s}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|MTH3002

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|Topology and physics

Although earlier moments, such as Euler’s use of graph theory to investigate the Konigsberg bridges problem (1736) could be singled out as the beginning of the study of topology, the subject only really became an important area of mathematics with the work of Poincare in the $1880 \mathrm{~s}$ and $1890 \mathrm{~s}$. While investigating the properties of solutions to differential equations, and especially problems in celestial mechanics, he was led to the study of smooth mappings between surfaces, to fixed points, singularities of vector fields, and other topics that would now be considered topological. He went on to give the first definitions of homotopy and homology and to lay the foundations of modern algebraic topology. Poincaré’s topological studies of solutions to differential equations as curves on manifolds was continued in the early 20 th century by Birkhoff and others, with the results eventually being systematically applied to mechanical systems by Kolmogorov, Arnold, and Moser. Simultaneously, other branches of the subject, such as differential topology and combinatorial topology began expanding, leading to a number of fixcd point theorems and to the clarification of useful concepts such as compactness, connectedness, and dimension.

Aspects of topology, then known as analysis situs or geometria situs, had made appearances in physics before this, of course. For example, Gauss’ law and Ampère’s law in electrodynamics are both topological in nature: they involve line or surface integrals that remain invariant under continuous deformations of the underlying curve or surface; in modern terminology, we would say that these integrals (the electric and magnetic fluxes) are topological invariants. In fact, integer linking numbers (chapter 5) made their first appearance in a study by Gauss of Ampère’s law.
Similarly, in fluid mechanics the study of vortices has a long history. Then, starting in the 1860s, Peter Tait and William Thomson (Lord Kelvin) tried to model atoms as knotted vortex lines in the ether. The motivations included the fact that the multiplicity of different atoms could be explained by the variety of different ways a vortex line could be knotted, and the fact that the stability of atoms could be attributed to the inability to untie a knot without cutting it open; in other words, atomic stability follows from topological stability of the knots. Different spectral lines could also be explained by different vibrational modes of the structure. The work of Tait and Kelvin led to knot theory becoming a major branch of topology, but after the idea of a space-filling ether was abandoned, knots disappeared from physics for almost a century, until they re-emerged in superstring theory and statistical mechanics, and then in other areas like optics.

## 数学代写|拓扑学代写Topology代考|Dirac monopoles

Although never seen experimentally, the possibility of isolated magnetic charges or monopoles has long been studied theoretically, starting with the work of Paul Dirac in the 1930 s [2]. In analogy to electric charges, a point-like magnetic monopole should produce a magnetic field (in SI units)
$$\boldsymbol{B}=\frac{\mu_{0} g}{4 \pi r^{2}} \hat{r}=-\nabla V(r),$$
where $g$ is the magnetic charge and $V=\mu_{0} g / 4 \pi r$ is the magnetic scalar potential. Because of the identity
$$\nabla^{2}\left(\frac{1}{r}\right)=-4 \pi \delta^{(3)}(\boldsymbol{r}),$$
the magnetic analog of Gauss’ law is
$$\nabla \cdot \boldsymbol{B}=g \mu_{0} \delta^{(3)}(\boldsymbol{r})$$
where $\delta^{(3)}(\boldsymbol{r})$ is the three-dimensional Dirac delta function and the magnetic charge density is $\rho_{m}(\boldsymbol{r})=g \delta^{(3)}(\boldsymbol{r})$.

Recall that when a particle of momentum $\boldsymbol{p}$ propagates with displacement $\boldsymbol{r}$, the wavefunction picks up a phase factor,
$$\psi \rightarrow \psi e^{i p \cdot r / \hbar}$$
The phase of a single wavefunction at a given point has no physical relevance, but the phase difference between points is meaningful, since it is measurable through interference effects. When there is a field present, the minimal coupling procedure of electromagnetism leads (for a particle of charge $e$ ) to an effective shifting of the momentum,
$$\boldsymbol{p} \rightarrow \boldsymbol{p}-{ }_{c}^{e} A .$$

## 数学代写|拓扑学代写Topology代考|Dirac monopoles

$$\boldsymbol{B}=\frac{\mu_{0} g}{4 \pi r^{2}} \hat{r}=-\nabla V(r),$$

$$\nabla^{2}\left(\frac{1}{r}\right)=-4 \pi \delta^{(3)}(\boldsymbol{r}),$$

$$\nabla \cdot \boldsymbol{B}=g \mu_{0} \delta^{(3)}(\boldsymbol{r})$$

$$\psi \rightarrow \psi e^{i p \cdot r / \hbar}$$

$$\boldsymbol{p} \rightarrow \boldsymbol{p}-{ }_{c}^{e} A .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。