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}}
\[We can see:
(i) the set $\{a\}$ is both open and closed;
(ii) the set $\{b, c\}$ is neither open nor closed;
(iii) the set $\{c, d\}$ is open but not closed;
(iv) the set $\{a, b, e, f\}$ is closed but not open.
In a discrete space every set is both open and closed, while in an indiscrete space$(X, \tau),$ all subsets of $X$ except $X$ and $\emptyset$ are neither open nor closed.\]
