### 数学代写|拓扑学代写Topology代考|MATH3061

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• 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 into a 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 ( $\mathbb{d}, \mathrm{d}$ ) where $\mathbb{T}$ is a set and $d$ is a distance function $d: \mathbb{I} \times \mathbb{T} \rightarrow \mathbb{R}$ 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 $p, q \in \mathbb{T}$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ 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 $B_{0}(c, r)={p \in \mathbb{T}: \mathrm{d}(p, c)<r}$. Metric balls definé a topology on a metric spacé.

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}

## 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.

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.

## 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 $p, q \in \mathbb{T}$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ 对所有人 $p, q, r \in \mathbb{T}$.
可以证明，上面的三个公理意味着 $\mathrm{d}(p, q) \geq 0$ 对于每一对 $p, q \in \mathbb{T}$. 在度量空间中 $\mathbb{T}$,一个带中心的开放公制 球 $c$ 和半径 $r$ 被定义为点集 $B_{0}(c, r)=p \in \mathbb{T}: \mathrm{d}(p, c)<r$. 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
$\mathrm{~ U l e f t { B _ { o } ( c , r )}$
一般拓扑空间的所有定义都适用于具有上述定义的拓扑的度量空间。但是，我们使用可能更直观的极限点概念给出 了替代定义。
正如我们已经提到的，拓扑的核心是连接一组点意味着什么的问题。毕竟，两个不同的点不能彼此相邻；它们只有 通过无数的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 $\mathrm{d}(\cdot, \cdot)$ 表示点集之间的最小距离 $P, Q \subseteq \mathbb{T}$
$$\mathrm{d}(p, Q)=\inf \mathrm{d}(p, q): q \in Q \mathrm{~d}(P, Q) \quad=\inf \mathrm{d}(p, q): p \in P, q \in Q$$

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