Metric space,open set

目录

• 引入：绝对值
• 度量空间
  • Example:
• 开集，闭集

引入：绝对值

distance\(:|a-b|\)
properties\(:(1)|x| \geq 0\),for all \(x \in R\),and \("=” \Leftrightarrow x=0\)
\((2):|a-b|=|b-a|(|x|=|-x|)\)
\((3):|x+y| \leq |x|+|y|\),for all \(x,y \in R\)
(\(|a-c| \leq |a-b|+|b-c|\))

度量空间

Distance function/metric space
Let \(X\) be a set.
\(\underline{Def:}\)A function \(X \times X \stackrel{d}{\longrightarrow}\mathbb{R}\)is called a distance function on \(X\)
1.\(\forall x,y\in X\),\(d(x,y)\geq 0\) and \("=” \Leftrightarrow x=y\)
2.\(\forall x,y\in X\),\(d(x,y)=d(y,x)\)
3.\(\forall x,y,z \in X\),\(d(x,z)\leq d(x,y)+d(y,z)\)

Example:

\(\mathfrak{A}:\)
1.\(x=(x_1,x_2,\dots,x_m),y=(y_1,y_2,\dots,y_m)\in \mathbb{R}^n\)
\(d_2(x,y):=\sqrt{|x_1-y_1|^2+\cdots+|x_m-y_m|^2}=|x-y|\)
\(d_2\) is a metric on \(\mathbb{R}^n\)(Cauchy inequality)
2.\(d_1(x,y):=|x_1-y_1|+|x_2-y_2|+\cdots+|x_m-y_m|\)
3.\(d_{\infty}(x,y)=max\{|x_1-y_1|,\dots,|x_m-y_m|\}\)
\(\mathfrak{B}:\)
X:a set.For \(x,y \in X\),let \[d(x,y):=\left\{
\begin{aligned}
1&if&x\leq y
\\
0&if&x =y
\end{aligned}
\right.
\]
\(d(x,y)\Rightarrow\)the discrete metric

开集，闭集

we may generalize the definitions about limits and convergence to metric space
\(\underline{Def}\) Let \((X,d)\) be a metric space,\(a_n(n \in \mathbb{N})\)be a seq in \(\mathrm{X}\).and \(\mathcal{L}\)in X
\(a_n(n \in \mathbb{N})\)converges to \(\mathcal{L}\)
(1)For \(r \geq 0\)and \(x_0 \in X\),we let \(B_r(x_0)=\{x \in X|d(x,x_0)\leq r\}\)(open ball)
(2).S is an open set(of\((X,d)\)),if \(\forall x \in S\),\(\exists r >0\)
（\(B_r(x_0)\subset S\)）open ball \(\Rightarrow\)open set
EX:
\((X,d):\)metric space.\(x_0 \in X,r \geq 0\)
Show that:(1)\(B_r(x_0)\)is open
(2)\(\{x \in X|d(x,x_0）> r\}\)is open
warning:A subset \(S\) of a topological space \((X, \mathcal{T})\) is said to be clopen if it is both open and closed in \((X, \mathcal{T})\)
Example. \(\quad\) Let \(X=\{a, b, c, d, e, f\}\) and
\[
\tau_{1}=\{X, \emptyset,\{a\},\{c, d\},\{a, c, d\},\{b, c, d, e, f\}\}
\]
We can see:

(i) the set \(\{a\}\) is both open and closed;

(ii) the set \(\{b, c\}\) is neither open nor closed;

(iii) the set \(\{c, d\}\) is open but not closed;

(iv) the set \(\{a, b, e, f\}\) is closed but not open.
In a discrete space every set is both open and closed, while in an indiscrete space\((X, \tau),\) all subsets of \(X\) except \(X\) and \(\emptyset\) are neither open nor closed.