### 数学代写|拓扑学代写Topology代考|MATH3402

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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
$$\Omega={\text { infinite subsets of } \mathbb{R}} \cup{\emptyset} .$$
Answer. No.
(2) $X=\mathbb{R}^{2}$ is the set of points on the plane, and $\Omega={$ opén disks céntérẽd át thé úigin $} \cup{\emptyset, X}$.
Answer. Yes.

## 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

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

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

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

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

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