Posted on Categories:Topology, 拓扑学, 数学代写

# 数学代写|拓扑学代写TOPOLOGY代考|Local Homeomorphisms and Sections

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|拓扑学代写TOPOLOGY代考|Local Homeomorphisms and Sections

Definition 12.1 A continuous map $f: X \rightarrow Y$ is a local homeomorphism if for every $x \in X$ there exist open sets $A \subset X, B \subset Y$ such that $x \in A, f(A)=B$ and the restriction $f: A \rightarrow B$ is a homeomorphism.
Example 12.2 Every continuous, 1-1 and open map is a local homeomorphism.
Lemma 12.3 Every local homeomorphism $f: X \rightarrow Y$ is open, and the fibres $f^{-1}(y), y \in Y$, are discrete.

Proof We want to show that the image $f(V)$ of an open set $V \subset X$ is a neighbourhood of each of its points. That is to say, for every $y \in f(V)$ there exists $U \subset Y$ open such that $y \in U \subset f(V)$.

Let $x \in V$ be such that $f(x)=y$; by assumption there are open sets $A \subset X, B \subset$ $Y$ such that $x \in A, f(A)=B$ and the restriction $f: A \rightarrow B$ is a homeomorphism. In particular $y \in f(V \cap A)$, and $U=f(V \cap A)$ is open in $B$ so also open in $Y$.
For every $y \in Y$ and $x \in f^{-1}(y)$ there exists an open neighbourhood $x \in A$ for which the restriction $f: A \rightarrow Y$ is $1-1$. Hence $f^{-1}(y) \cap A={x}$, proving that the subspace topology on the fibres $f^{-1}(y)$ is discrete.

If $p: X \rightarrow Y$ is a map between sets, a function $s: Y \rightarrow X$ is called a section of $p$ if $p(s(y))=y$ for every $y \in Y$. A necessary condition for $p$ to have a section is that $p$ be onto; vice versa, the axiom of choice says exactly that any map admits sections.

In contrast-moving back to the topological world-continuous sections of continuous surjective maps do not exist, in general.

## 数学代写|拓扑学代写TOPOLOGY代考|Covering Spaces

Definition 12.5 Let $X$ be a connected space. A space $E$ together with a continuous map $p: E \rightarrow X$ is a covering space of $X$ if every point $x \in X$ is contained in an open set $V \subset X$ whose pre-image $p^{-1}(V)$ is the disjoint union of open sets $U_i$ with the property that $p: U_i \rightarrow V$ is a homeomorphism for every $i$.

The space $X$ is called the base (space) of the covering space, $E$ is the total space and $p$ is the covering map. The sets $p^{-1}(x), x \in X$, are called fibres of the covering space.

An open set $V \subset X$ is an admissible (open) set of the covering $p$ if it fulfils the condition of Definition 12.5. With other words $V \subset X$ is admissible if we can write $p^{-1}(V)=\cup_i U_i$, where:

1. every $U_i$ is open in $E$ and the restrictions $p: U_i \rightarrow V$ are homeomorphisms;
2. $U_i \cap U_j=\emptyset$ for every $i \neq j$.
Clearly, an open set contained in an admissible open set is still admissible. We will say that the covering space is trivial if the whole base $X$ is an admissible set.

If $p: E \rightarrow X$ is a covering space, from the definition every point $e \in E$ has an open neighbourhood homeomorphic to an open neighbourhood of $p(e)$. For later use we note that this implies that if $X$ is locally path connected, also $E$ is locally path connected.

Definition 12.6 A covering space $p: E \rightarrow X$ is connected if the total space $E$ is connected.

Example 12.7 Let $X$ be connected and $F$ a non-empty discrete space. The projection on the first factor $X \times F \rightarrow X$ is a trivial covering space, and it’s connected if and only if $F$ consists of one point.

## 数学代写|拓扑学代写TOPOLOGY代考|Covering Spaces

$U_i \cap U_j=\emptyset$ 对于每个$i \neq j$。

## 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。