度量空间，开集，闭集

@

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:

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.\(d_{\infty}(x,y)=max\{|x_1-y_1|,\dots,|x_m-y_m|\}\)

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 X.and \(\mathbb{L}\)in X
\(a_n(n \in \mathbb{N})\)converges to L
(1)For \(r \geq 0\)and \(x_0 \in X\),we let \(B_r(x_0)=\{x \in X|d(x,x_0)\leq r\}\)