度量空间的一个例子:离散度量空间

Let $X$ be any nonempty set.For any $x,y\in X$,define

$d(x,y)=1$  if $x\neq y$

$d(x,y)=0$  if $x=y$.

Then $(X,d)$ is a metric space.The metric $d$ is called discret metric and the space $(X,d)$ is called discret metric space.

Proof:

(1)For any given $x\in X$,$d(x,x)=0$.
(2)For any $x,y\in X$,$d(x,y)=0$,or $d(x,y)=1$.So $d(x,y)\geq 0$.
(3)If $x=y$,then $y=x$.And if $x\neq y$,$y\neq x$.So $d(x,y)=d(y,x)$
(4)If $x=y$,then it is easy to verify that $d(x,y)\leq d(x,z)+d(z,y)$.If $x\neq y$,then $x=z$ and $y=z$ can not be hold at the same time.So $d(x,y)\leq d(x,z)+d(z,y)$.

 

 

注:这个例子无非就是：一个非空集合里的元素到自身的距离都是0，而不同元素之间的距离都是1.这个例子告诉我们，可以在任意非空集合上定义一个度量.

转载于:https://www.cnblogs.com/yeluqing/archive/2013/02/22/3827456.html