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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|Selected Solutions 4

4.2) The Cantor set.
4.3) (a) No. For instance, take $\Omega_{1}$ to be indiscrete and take $\Omega_{2}$ to be discrete.
(b) Yes. Assume that $\left(X, \Omega_{1}\right)$ is disconnected. Then $\left(X, \Omega_{1}\right)$ has a nonempty proper clopen subset, say, $A$. Since
$$\Omega_{1} \subseteq \Omega_{2},$$
the set $A$ is clopen in $\Omega_{2}$. Hence $\left(X, \Omega_{2}\right)$ is disconnected.
4.4) (a) Any subset. Assume that $A \subseteq[0, \infty)$ is disconnected. By Theorem $4.4$, there exists a continuous surjection
$$f: A \rightarrow S^{0} \text {. }$$
Note that any open set in $A$ consists of the elements of $A$ that are larger than some number $M$, and that any closed set in $A$ consists of the elements of $A$ that are smaller than or equal to some number $m$. Therefore, the clopen set $f^{-1}(1)$ in $A$ has to be either the empty set or the whole set $A$, contradicting that $f$ is surjective.
(b) Any infinite subset. On one hand, any non-singleton finite set $A$ is the union of two nonempty finite subsets. Note that a subset is finite if and only if it is closed. By Theorem 4.4, we infer that $A$ is disconnected. On the other hand, assume to the contrary that an infinite set $A$ is disconnected. By Theorem $4.4$, there exists a continuous surjection
$$f: A \rightarrow S^{0} \text {. }$$
Then each of the preimages $f^{-1}(\pm 1)$ is closed in $A$. Hence the set
$$A-f^{-1}(-1) \cup f^{-1}(1)$$

## 数学代写|拓扑学代写Topology代考|Separation and Countability Axioms

In this chapter, we consider natural restrictions on the topology of a space that measures how far away the space is from a metrizable space. The restrictions are called Separation Axioms. As will be seen, metric spaces satisfy all of the separation axioms. Topologists assigned the classes of spaces satisfying different axioms the names $T_{1}, T_{2}, T_{3}$, and $T_{4}$ spaces, respectively. Later this system of numbering was extended to include $T_{0}, T_{e}, T_{\pi}, T_{5}$, and $T_{6}$. The idea was supposed to be that every $T_{i}$ space is a special kind of $T_{j}$ space if $i>j$. But this is not necessarily true, as commonly accepted definitions vary. For example, a $T_{3}$ space may do not satisfy the $T_{2}$ axiom.
In Section $5.1$ we introduce topological spaces satisfying the $T_{i}$ axiom for each $i=0,1,2,3,4$. We further study Hausdorff spaces in Section $5.2$ and regular and normal spaces in Section 5.3. We end this chapter by discussing common countability axioms.

Let $f, g: X \rightarrow Y$ be maps between sets. The coincidence set of $f$ and $g$ is the set
$$C(f, g)={x \in X: f(x)=g(x)} .$$
5.1. Axioms $T_{0}, T_{1}, T_{2}, T_{3}$, and $T_{4}$
Definition 5.1. Two points $p$ and $q$ in a topological space are topologically indistinguishable if they have exactly the same neighborhoods, and topologically distinguishable otherwise. Two sets in a topological space are separated by neighborhoods if they have disjoint neighborhoods. A separation axiom is a condition that a topological space may satisfy.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Selected Solutions 4

4.2）康托集。
4.3) (a) 不。例如，采取 $\Omega_{1}$ 轻率并采取 $\Omega_{2}$ 是离散的。
(b) 是的。假使，假设 $\left(X, \Omega_{1}\right)$ 已断开连接。然后 $\left(X, \Omega_{1}\right)$ 有一个非空的真 cloopen 子集，比如说， $A$. 自从
$$\Omega_{1} \subseteq \Omega_{2},$$

4.4) (a) 任何子集。假使，假设 $A \subseteq[0, \infty)$ 已断开连接。按定理4.4, 存在一个连续的满射
$$f: A \rightarrow S^{0} \text {. }$$

(b) 任何无限子集。一方面，任何非单例有限集 $A$ 是两个非空有限子集的并集。请注意，子集是有限的当且仅当它是封闭的。根据定理 4.4，我们推断 $A$ 已断开连接。 另一方面，相反地假设一个无限集 $A$ 已断开连接。按定理 $4.4$, 存在一个连续的满射
$$f: A \rightarrow S^{0} .$$

$$A-f^{-1}(-1) \cup f^{-1}(1)$$

## 数学代写|拓扑学代写Topology代考|Separation and Countability Axioms

$$C(f, g)=x \in X: f(x)=g(x) .$$
5.1。公理 $T_{0}, T_{1}, T_{2}, T_{3}$ ，和 $T_{4}$

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