度量空間的乙個例子 離散度量空間

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:這個例子無非就是：乙個非空集合裡的元素到自身的距離都是0，而不同元素之間的距離都是1.這個例子告訴我們，可以在任意非空集合上定義乙個度量.

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