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# 经济代写|Open and closed sets.微观经济学代写

## 经济代写

DEFINITION IV.4 (INTERIOR POINT). $x^{}$ is called an interior point of some set $M$ if there exists an $\varepsilon$-ball $K$ with center $x^{}$ such that $K \subseteq M$.
EXAMPLE IV.2. Point 1 is not an interior point of $[0,1]$.
EXAMPLE IV.3. $x^{}$ is an interior point of the $\varepsilon$-ball $K$ with center $x^{}$.
DEFINITION IV.5 (OPEN SET). A set that consists of interior points only is called open. The empty set (symbol $\emptyset$ ) is open.

A set $M$ is open if you can take an arbitrary point of this set and find an $\varepsilon$-ball $K$ that is contained in $M$.
EXAMPLE IV.4. In $\mathbb{R}^{1},(0,1)={x \in \mathbb{R}: 0<x<1}$ is an open set. $\mathbb{R}^{\ell}$ is open. Every $\varepsilon$-ball is an open set. You can see this by sketching the $\varepsilon$-ball and by finding a second, smaller one, around every point in the $\varepsilon$-ball (see figure IV.S).

Definition IV.6 (COMPLEMENT). $\mathbb{R}^{\ell} \backslash M=\left{x \in \mathbb{R}^{\ell}: x \notin M\right}$ is the complement of $M$ (in $\left.\mathbb{R}^{\ell}\right)$.
DEFINITION IV.7 (BOUNDARY POINT). $x^{}$ is called boundary point of a set $M$ if for all $\varepsilon$-balls $K$ with center $x^{}$ the following two conditions are fulfilled:
\begin{aligned} K \cap M & \neq \emptyset \text { and } \ K \cap\left(\mathbb{R}^{\ell} \backslash M\right) & \neq \emptyset . \end{aligned}
Figure IV.4 illustrates the definition of a boundary point in two-dimensional space.
EXERCISE IV.4. Find the boundary points of $(0,1) \subset \mathbb{R}$, of $[0,1]$, and of $[0,0]$ !

LEMMA IV.1. Assume that $x^{}$ is contained in $M$. Then, $x^{}$ is an interior point of $M$ if and only if $x^{*}$ is not a boundary point of $M$.

1. THE VECTOR SPACE OF GOODS AND ITS TOPOLOGY
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FIGURE IV.4. A boundary point
PROOF. If $x^{}$ is an interior point of $M$, we can find an $\varepsilon$-ball $K$ with center $x^{}$ such that $K \subseteq M$. Then, $K \cap\left(\mathbb{R}^{\ell} \backslash M\right)=\emptyset$ and $x^{}$ is not a boundary point of $M$. If $x^{}$ is not an interior point of $M$, every e-ball $K$ with center $x^{}$ fulfills $K \nsubseteq M$ or, differently put, $K \cap\left(\mathbb{R}^{\ell} \backslash M\right) \neq \emptyset$. Also, every $\varepsilon$-balls $K$ with center $x^{}$ contains $x^{}$ Therefore, $x^{}$ is contained in both $K$ and $M$ so that $K \cap M \neq \emptyset$ holds. Thus, $x^{*}$ is a boundary point of $\operatorname{set} M$.

DEFINITION IV.8 (CLOSED SET). A set containing all its boundary points is called closed. $\mathbb{R}^{\ell}$ itself is also closed.
EXAMPLE IV.5. In $\mathbb{R}^{1},[0,1]={x \in \mathbb{R}: 0 \leq x \leq 1}$ is a closed set.
LEMMA IV.2. The complement of an open set is a closed set. The complement of a closed set is an open set.

It can be the case that a set is neither closed nor open. Consider, for example, the set ${0} \cup(1,2)$. On the other hand, $\mathbb{R}^{\ell}$ and $\emptyset$ are both open and closed.

DEFINITION IV.9 (COMPACT SET).A set $M \subseteq \mathbb{R}^{\ell}$ is called compact if it is closed and bounded.

EXAMPLE IV.6. $[0,1]$ is compact. $\mathbb{R}^{\ell}$ is closed but not bounded. $\varepsilon$-balls are bounded but not closed. Hence, neither $\mathbb{R}^{\ell}$ nor $\varepsilon$-balls are compact.
1.4. Sequences and convergence. It is often helpful to consider the above concepts from the point of view of sequences in $\mathbb{R}^{l}$.

$$\开始{对齐} K \cap M & \neq \emptyset \text { 和 } \ K \cap\left(\mathbb{R}^{\ell} \backslash M\right) & \neq \emptyset 。 \end{对齐}$$

1. 货物的向量空间及其拓扑
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图 IV.4。边界点
证明。如果$x^{}$ 是$M$ 的内点，我们可以找到一个中心为$x^{}$ 的$\varepsilon$-球$K$，使得$K \subseteq M$。则$K \cap\left(\mathbb{R}^{\ell} \backslash M\right)=\emptyset$ 且$x^{}$ 不是$M$ 的边界点。如果 $x^{}$ 不是 $M$ 的内点，则每个中心为 $x^{}$ 的电子球 $K$ 满足 $K \nsubseteq M$ 或者，换句话说，$K \cap \left(\mathbb{R}^{\ell} \backslash M\right) \neq \emptyset$.此外，每个中心为 $x^{}$ 的 $\varepsilon$-balls $K$ 包含 $x^{}$ 因此，$x^{}$ 包含在 $K$ 和 $M$ 中，所以$K \cap M \neq \emptyset$ 成立。因此，$x^{*}$ 是 $\operatorname{set} M$ 的边界点。

1.4.序列和收敛。从 $\mathbb{R}^{l}$ 中的序列的角度考虑上述概念通常很有帮助。

## 经济代考

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## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码

## 复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。