## 澳洲代写｜BFW3540 ｜Modelling in finance金融建模 蒙纳士大学

statistics-labTM为您提供蒙纳士大学（Monash University）Modelling in finance金融建模澳洲代写代考辅导服务！

Topics include the development and application of financial spreadsheets, Excel and Visual Basic programming in financial modelling, modelling company financial statements, fixed income securities analysis, asset allocation and portfolio analysis, optimization using Solver, Interest rate models, option pricing models, numerical methods and risk management models.

## Fourier Transform傅立叶变换案例

In this section, we recall the Fourier transform definition, both for notational reasons and for the reader’s convenience.
The Fourier transform, for $f \in \mathcal{S}\left(\mathbb{R}^m\right)$, is denoted here as
$$\widehat{f}(\xi):=\mathcal{F}f:=\int_{\mathbb{R}^m} \mathrm{e}^{i \xi x} f(x) \mathrm{d} x,$$
where $\mathcal{S}\left(\mathbb{R}^m\right)$ is the Schwartz space of $\mathcal{C}^{\infty}\left(\mathbb{R}^m\right)$ functions of rapid decrease, see [RS75]. This is not the usual definition found in the mathematical literature. However, it is standard in probability, see [Chu01] and in the finance literature, see [CT04].

The Fourier transform is a linear bijection from $\mathcal{S}\left(\mathbb{R}^m\right)$ onto $\mathcal{S}\left(\mathbb{R}^m\right)$, whose inverse is given by the Fourier inversion formula

f(x)=\mathcal{F}^{-1}\widehat{f}=\frac{1}{(2 \pi)^m} \int_{\mathbb{R}^m} \mathrm{e}^{-i \xi x} \widehat{f}(\xi) \mathrm{d} \xi

We also recall the Fourier transform for $f \in \mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$, where $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ is the space of tempered distributions, which is the dual of $\mathcal{S}\left(\mathbb{R}^m\right)$, the Fourier transform can be defined as
$$(\mathcal{F}[f], \varphi)=(2 \pi)^m\left(f, \mathcal{F}^{-1}[\varphi]\right) \quad \varphi \in \mathcal{S}\left(\mathbb{R}^m\right),$$
see [RR04]. This definition makes the Fourier transform in $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ an extension of the Fourier transform in $\mathcal{S}\left(\mathbb{R}^m\right)$. The Fourier transform for $L^1\left(\mathbb{R}^m\right)$ and $L^2\left(\mathbb{R}^m\right)$ are restrictions of the Fourier transform for $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$.

The Fourier transform has several useful properties. Some of them are reviewed below with the purpose of calling attention to the notation used here:

• $\mathcal{F}f(x-a)=\mathrm{e}^{i a \xi} \widehat{f}(\xi)$
• $D^\alpha \widehat{f}(\xi)=\mathcal{F}\left(i x)^\alpha f\right$
• $(-i \xi)^\alpha \widehat{f}(\xi)=\mathcal{F}\leftD^\alpha f\right$
Some specific distributions are often used in this thesis. To present the notation, we give a brief overview of them. First, consider the Cauchy principal value
\begin{aligned} 1 / x: \mathcal{S}(\mathbb{R}) & \rightarrow \mathbb{R} \ f & \rightarrow(1 / x, f):=f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x, \end{aligned}
where
$$f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x:=\lim {\epsilon \downarrow 0}\left(\int\epsilon^{\infty} \frac{f(x)}{x} \mathrm{~d} x+\int_{-\infty}^{-\epsilon} \frac{f(x)}{x} \mathrm{~d} x\right) .$$
This defines a distribution in $\mathcal{S}^{\prime}(\mathbb{R})$.

## Probability and Stochastic Processes概率与随机过程案例

In this section, we present a brief overview of the topics on probability and stochastic processes used herein. References on the subject are [CW90] and [Sat99].

In this thesis, the triple $(\Omega, \mathcal{F}, \mathbb{P})$ denotes a complete probability space, where $\Omega$ is a set of points $\omega, \mathcal{F}$ is a $\sigma$-algebra containing all $\mathbb{P}$-null sets, and $\mathbb{P}$ is a probability measure. When we say that $X$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, we mean that $X$ is real-valued function on $\Omega$, measurable with respect to $\mathcal{F}$.
The characteristic function of a random variable is defined as
$$\varphi(z)=\mathbb{E}\left[\mathrm{e}^{i z X}\right] .$$
For properties of the characteristic function and a review of probability theory we refer to [CW90].

The filtered complete probability space is denoted by $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$, where, as in [Pro04], we write $\mathbb{F}$ for the filtration $\left(\mathcal{F}t\right){0 \leq t \leq \infty}$ and we assume that $\mathcal{F}_0$ contains all the $\mathbb{P}$-null sets. We use $\stackrel{\mathbb{P}}{\longrightarrow}$ to denote convergence in probability, see [CW90] and the French acronyms càdlàg (continu à droite, limité à gauche) is used to define the right continuous, left limited process, see [Pro04].

The main class of stochastic processes we are interested in this work are the Levy processes, see [Sat99] for a comprehensive treatment of the subject. We briefly review the definition of a Levy process
Definition: A Levy process is a càdlàg stochastic process, $\left(X_t\right)_{t \geq 0}$, on $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ taking values in $\mathbb{R}$ and with the following properties:

• Independent increments. That is, given $t_0 \leq \ldots \leq t_N$, and defined $Y_n:=X_{t_n}-X_{t_{n-1}}$ we have $\left{Y_n\right}_{n=1}^N$ independent;
• Stationary increments. That is, the distribution of $X_{t+s}-X_t$ does not depend on $t$;
• Stochastic continuity. That is,
$$X_{t+h} \underset{h \downarrow 0}{\stackrel{\mathbb{P}}{\longrightarrow}} X_t$$
An important stochastic process is the Brownian motion.

## 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。

## 澳洲代写｜MTH3130 ｜Topology: The mathematics of shape拓扑学形状数学 蒙纳士大学

statistics-labTM为您提供蒙纳士大学（Monash University）Topology: The mathematics of shape拓扑学形状数学澳洲代写代考辅导服务！

From point-set topology to manifolds: sets, topological spaces, basis of topology, and properties of spaces such as compact, connected, and Hausdorff. Maps between spaces and their properties, including continuity, homeomorphism, and homotopy.

Constructing spaces via subspace, product, identification, and cell complexes. Manifolds. Additional topics from algebraic and low-dimensional topology may include fundamental group and Seifert-van Kampen theorem, classification of surfaces, and topics in knot theory. Throughout, examples of spaces will include Euclidean spaces, surfaces (real projective plane, Klein bottle, Mobius strip), complexes, function spaces, and others.

## Topology拓扑学问题集

Show that if $f: X \rightarrow Y$ induces an isomorphism in homology with coefficients in the prime fields $\mathbb{F}_p$ (for all primes $p$ ) and $\mathbb{Q}$, then it induces an isomorphism in homology with coefficients in $\mathbb{Z}$. (Hint: 18 (c).)

Let $A \subseteq X$ and $B \subseteq Y$ be subsets. Construct a natural chain map
$$S_(X, A) \otimes S_(Y, B) \rightarrow S_(X \times Y, A \times Y \cup X \times B)$$ that is a homology isomorphism if $A$ and $B$ are open. (Hint: Problem 26., or its proof, might be useful.) So there is a natural “relative cross product” map $$H_(X, A ; R) \otimes_R H_(Y, B ; R) \rightarrow H_(X \times Y, A \times Y \cup X \times B ; R)$$
that is an isomorphism if $A$ and $B$ are open, $R$ is a PID, and either $H_(X, A ; R)$ or $H_(Y, B ; R)$ is free over $R$.

(a) What is the $k$ th Betti number of $\left(S^1\right)^n$ ?
(b) Define an equivalence relation on $\mathbb{R}^n$ by saying that two vectors are equivalent if they differ by a vector with entries in $\mathbb{Z}$. Identify the quotient space of $\mathbb{R}^n$ by this equivalence relation with the product space $\left(S^1\right)^n$. Let $M$ be an $n \times n$ matrix with entries in $\mathbb{Z}$. It defines a linear map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ in the usual way. Show that this map descends to a self-map of $\left(S^1\right)^n$. Compute the effect of this map on $H_n\left(\left(S^1\right)^n\right)$.

## 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|MATH421

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。

## 数学代写|拓扑学代写Topology代考|Continuity and Convergence of Nets

One can, of course, define convergence for sequences in topological spaces as for metric spaces.

Definition 3.2.1. Let $(X, \mathcal{T})$ be a topological space. A sequence $\left(x_n\right)_{n=1}^{\infty}$ in $X$ is said to converge to $x \in X$ if, for each $N \in \mathcal{N}_x$, there is $n_N \in \mathbb{N}$ such that $x_n \in N$ for all $n \geq n_N$.

This definition is perfectly fine, but if one attempts to prove analogues of results for convergent sequences in metric spaces, problems show up. Proposition 2.3.4, for example, is no longer true in general topological spaces.

Example 3.2.2. Let $X$ be an uncountable set equipped with the topology of Example 3.1.2(e); that is, the open sets are $\varnothing$ and those with a countable complement. Fix a point $x_0 \in X$. Then $X \backslash\left{x_0\right}$ is not closed, so that $\overline{X \backslash\left{x_0\right}}=X$ must hold. Let $\left(x_n\right){n=1}^{\infty}$ be a sequence in $X \backslash\left{x_0\right}$, and let $U:=X \backslash\left{x_1, x_2, \ldots\right}$. Due to the nature of our topology, $U$ is open and thus is a neighborhood of $x_0$. However, $x_n \notin U$ for all $n \in \mathbb{N}$ by definition, so that $\left(x_n\right){n=1}^{\infty}$ cannot converge to $x_0$.

A less contrived example for the failure of Proposition 2.3.4 in general topological spaces is given in Exercise 11 below.

So, how are we going to define continuity on arbitrary topological spaces? Of course, we could try it via sequences as for metric spaces, but in view of Example 3.2.2, we are likely to run into unexpected difficulties. Of the four equivalent conditions of Theorem 2.3.7, the fourth one doesn’t make any explicit reference to a metric. We thus use it as the definition of continuity.

## 数学代写|拓扑学代写Topology代考|Compactness

Compactness for topological spaces is defined as in the metric situation.
Definition 3.3.1. Let $(X, \mathcal{T})$ be a topological space, and let $S \subset X$. An open cover for $S$ is a collection $\mathcal{U}$ of open subsets of $X$ such that $S \subset \bigcup{U: U \in \mathcal{U}}$.
Definition 3.3.2. A subset $K$ of a topological space $(X, \mathcal{T})$ is called compact if, for each open cover $\mathcal{U}$ of $K$, there are $U_1, \ldots, U_n \in \mathcal{U}$ such that $K \subset$ $U_1 \cup \cdots \cup U_n$.

Before we flesh out this definition with examples (nonmetrizable ones), we introduce yet another definition.

Definition 3.3.3. A topological space $(X, \mathcal{T})$ has the finite intersection property if, for any collection $\mathcal{F}$ of closed subsets of $X$ such that $\bigcap{F: F \in \mathcal{F}}=$ $\varnothing$, there are $F_1, \ldots, F_n \in \mathcal{F}$ such that $F_1 \cap \cdots \cap F_n=\varnothing$.
The following is straightforward (just pass to complements).
Proposition 3.3.4. Let $(X, \mathcal{T})$ be a topological space. Then the following are equivalent.
(i) $X$ is compact.
(ii) $X$ has the finite intersection property.
The reason why we introduced the finite intersection property at all is that it is sometimes easier to verify than compactness.

Example 3.3.5. Let $R$ be a commutative ring with identity. We claim that $\operatorname{Spec}(R)$ has the finite intersection property (and thus is compact). Let $\mathcal{I}$ be a family of ideals of $R$ such that
$$\bigcap{V(I): I \in \mathcal{I}}=V\left(\sum{I: I \in \mathcal{I}}\right)=\varnothing .$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Continuity and Convergence of Nets

3.2.1.定义设$(X, \mathcal{T})$为拓扑空间。如果对于每一个$N \in \mathcal{N}x$，都有一个$n_N \in \mathbb{N}$使得对于所有$n \geq n_N$，都有一个$x_n \in N$，那么我们就说$X$中的一个序列$\left(x_n\right){n=1}^{\infty}$收敛到$x \in X$。

## 数学代写|拓扑学代写Topology代考|Compactness

3.3.1.定义设$(X, \mathcal{T})$为拓扑空间，设$S \subset X$。$S$的开放覆盖是$X$的开放子集的集合$\mathcal{U}$，例如$S \subset \bigcup{U: U \in \mathcal{U}}$。
3.3.2.定义拓扑空间$(X, \mathcal{T})$的子集$K$被称为紧化，如果对于$K$的每个开盖$\mathcal{U}$，存在$U_1, \ldots, U_n \in \mathcal{U}$使得$K \subset$$U_1 \cup \cdots \cup U_n。 在用示例(不可度量的示例)充实这个定义之前，我们先介绍另一个定义。 3.3.3.定义拓扑空间(X, \mathcal{T})具有有限交性质，如果对于X的闭子集的任何集合\mathcal{F}使得\bigcap{F: F \in \mathcal{F}}=$$\varnothing$，存在$F_1, \ldots, F_n \in \mathcal{F}$使得$F_1 \cap \cdots \cap F_n=\varnothing$。

(i) $X$是紧凑的。
(ii) $X$具有有限交性质。

$$\bigcap{V(I): I \in \mathcal{I}}=V\left(\sum{I: I \in \mathcal{I}}\right)=\varnothing .$$

## 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|MATH622

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。

## 数学代写|拓扑学代写Topology代考|Compactness for Metric Spaces

The notion of compactness is one of the most crucial in all of topology (and one of the hardest to grasp).

Definition 2.5.1. Let $(X, d)$ be a metric space, and let $S \subset X$. An open cover for $S$ is a collection $\mathcal{U}$ of open subsets of $X$ such that $S \subset \cup{U: U \in \mathcal{U}}$.
Definition 2.5.2. A subset $K$ of a metric space $(X, d)$ is called compact if, for each open cover $\mathcal{U}$ of $K$, there are $U_1, \ldots, U_n \in \mathcal{U}$ such that $K \subset U_1 \cup$ $\cdots \cup U_n$.

Definition 2.5.2 is often worded as, “A set is compact if and only if each open cover has a finite subcover.”

Examples 2.5.3. (a) Let $(X, d)$ be a metric space, and let $S \subset X$ be finite; that is, $S=\left{x_1, \ldots, x_n\right}$. Let $\mathcal{U}$ be an open cover of $X$. Then, for each $j=1, \ldots, n$, there is $U_j \in \mathcal{U}$ such that $x_j \in U_j$. It follows that $S \subset$ $U_1 \cup \cdots \cup U_n$. Hence, $S$ is compact.
(b) Let $(X, d)$ be a compact metric space, and let $\varnothing \neq K \subset X$ be compact. Fix $x_0 \in K$. Since $\left{B_r\left(x_0\right): r>0\right}$ is an open cover of $K$, there are $r_1, \ldots, r_n>0$ such that
$$K \subset B_{r_1}\left(x_0\right) \cup \cdots \cup B_{r_n}\left(x_0\right) .$$
With $R:=\max \left{r_1, \ldots, r_n\right}$, we see that $K \subset B_R\left(x_0\right)$, so that $\operatorname{diam}(K) \leq$ $2 R<\infty$. This means, for example, that any unbounded subset of $\mathbb{R}^n$ (or, more generally, of any normed space) cannot be compact. In particular, the only compact normed space is ${0}$.
(c) Let $X=(0,1)$ be equipped with the usual metric. For $r \in(0,1)$, let $U_r:=(r, 1)$. Then $\left{U_r: r \in(0,1)\right}$ is an open cover for $(0,1)$ which has no finite subcover.

Before we turn to more (and more interesting) examples of compact metric spaces, we establish a few hereditary properties.

## 数学代写|拓扑学代写Topology代考|Topological Spaces-Definitions and Examples

A topological space is supposed to be a set that has just enough structure to meaningfully speak of continuous functions on it. In view of Corollary 2.3.10, a reasonable approach would be to axiomatize the notion of an open set:
Definition 3.1.1. Let $X$ be a set. $A$ topology on $X$ is a subset $\mathcal{T}$ of $\mathfrak{P}(X)$ such that:
(a) $\varnothing, X \in \mathcal{T}$;
(b) If $\mathcal{U} \subset \mathcal{T}$ is arbitrary, then $\bigcup{U: U \in \mathcal{U}}$ lies in $\mathcal{T}$;
(c) If $U_1, U_2 \in \mathcal{T}$, then $U_1 \cap U_2 \in \mathcal{T}$.
The sets in $\mathcal{T}$ are called open. A set together with a topology is called a topological space.

We often write $(X, \mathcal{T})$ for a topological space $X$ with topology $\mathcal{T}$; sometimes, if the topology is obvious or irrelevant, we may also simply write $X$.
Examples 3.1.2. (a) Let $(X, d)$ be a metric space, and let $\mathcal{T}$ denote the collection of all subsets of $X$ that are open in the sense of Definition 2.2.3. By Proposition 2.2.5, $\mathcal{T}$ is indeed a topology. It is clear that $\mathcal{T}$ does not depend on the particular metric $d$, but only on its equivalence class: any metric on $X$ equivalent to $d$ yields the same topology. Topological spaces of this type are called metrizable.
(b) Let $X$ be any set, and let $\mathcal{T}=\mathfrak{P}(X)$. This is just a special case of the first example: equip $X$ with the discrete metric. Such topological spaces are called discrete.
(c) Let $X$ be any set, and let $\mathcal{T}={\varnothing, X}$. Such topological spaces are called chaotic.
(d) Let $X$ be any set, and let $\mathcal{T}$ consist of $\varnothing$ and all subsets of $X$ with finite complement.
(e) Let $X$ be any set, and let $\mathcal{T}$ consist of $\varnothing$ and all subsets of $X$ with countable complement.
(f) Let $(X, \mathcal{T})$ be a topological space, and let $Y \subset X$. The relative topology on $Y$ (or the topology inherited from $X$ ) is the collection
$$\left.\mathcal{T}\right|_Y:={Y \cap U: U \in \mathcal{T}}$$
of subsets of $Y$. It is clearly a topology on $Y$. The space $\left(Y,\left.\mathcal{T}\right|_Y\right)$ is then called a subspace of $X$.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Compactness for Metric Spaces

2.5.1.定义设$(X, d)$是度量空间，设$S \subset X$。$S$的开放覆盖是$X$的开放子集的集合$\mathcal{U}$，例如$S \subset \cup{U: U \in \mathcal{U}}$。
2.5.2.定义度量空间$(X, d)$的子集$K$被称为紧化，如果对于$K$的每个开盖$\mathcal{U}$，存在$U_1, \ldots, U_n \in \mathcal{U}$使得$K \subset U_1 \cup$$\cdots \cup U_n。 定义2.5.2通常被表述为:“一个集合是紧的当且仅当每个开盖都有一个有限的子盖。” 例2.5.3。(a)设(X, d)为度量空间，S \subset X为有限空间;也就是S=\left{x_1, \ldots, x_n\right}。让\mathcal{U}成为X的公开封面。然后，对于每个j=1, \ldots, n，有U_j \in \mathcal{U}使得x_j \in U_j。由此可知S \subset$$U_1 \cup \cdots \cup U_n$。因此，$S$是紧凑的。
(b)设$(X, d)$为紧致度量空间，设$\varnothing \neq K \subset X$为紧致度量空间。修复$x_0 \in K$。因为$\left{B_r\left(x_0\right): r>0\right}$是$K$的公开封面，所以有$r_1, \ldots, r_n>0$
$$K \subset B_{r_1}\left(x_0\right) \cup \cdots \cup B_{r_n}\left(x_0\right) .$$

$$f_i^* u_0=f_i^\left(P D[P]_{\mathbb{C} P^2}\right)=P D\left[f_i^{-1}(P)\right]_M=P D\left[L_i\right]_M$$与$i=0$或1相同，并且来自标识\begin{aligned} {\left[M, \mathbb{C} P^2\right] } & \stackrel{\cong}{ } H^2(M, \mathbb{Z}) \ {[f] } & \longmapsto f^ u_0 \end{aligned}

## 数学代写|拓扑学代写Topology代考|Framed cobordisms

$$\partial S=L_1 \sqcup \bar{L}_0 \subset{1} \times M \sqcup{0} \times M .$$
$D^2$是一个嵌入在$S$中的小圆盘，$M$中的$L_1$和$L_2$的框架扩展到$[0,1] \times M$中的$S \backslash D^2$的框架(因为$S \backslash D^2$的变形缩回到包含$L_0$和$L_1$的一维复合体，在这样一个复合体上，任何可定向的2平面束都是微不足道的)。现在我们在$[0,1] \times M$中嵌入一个圆柱体$[0,1] \times S^1$，这样
$$\begin{gathered} {[0,1] \times S^1 \cap{0} \times M=\emptyset,} \ {[0,1] \times S^1 \cap{1} \times M={1} \times S^1,} \end{gathered}$$

$$[0,1] \times S^1 \cap\left(S \backslash \operatorname{Int}\left(D^2\right)\right)={0} \times S^1=\partial D^2,$$

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## MATLAB代写

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