metric spaces

In an arbitrary set, we are able to set some relations between the elements.

The order relationship in real number system is one of them.

[31] For each pair of real numbers $a$ and $b$, exactly one of the following is true:
       $a=b$,  $a<b$,  or  $b<a$
[32] If $a<b$ and $b<c$, then $a<c$
[33] If $a<b$, then $a+c<b+c$
[34] If $a<b$, then $ac<bc$, whenever $0<c$

Now we will introduce a new relation. For any elements $a, b$ in an arbitrary set $A$, there is a function $d(x,y)\in \mathbb{R}$ which satisfies following conditions.

(1)$d(a,b)\geq 0, \quad (d(a,b)=0 \quad \mbox{if and only if}  a=b)$
(2)$d(a,b)=d(b,a)$
(3)$d(a,b)+d(b,c)\geq d(a,c), \quad (\mbox{for any}   c\in A)$

Then, we call the set $A$ a metric space. (3) is the triangle inequality. The function $d$ is namely a distance function. The metric space is a set which has a quantifiable distance between elements.

For example, $d(a,b)=|b-a|, (a,b\in\mathbb{R})$ is a distance function. Therefore, $\mathbb{R}$ becomes a metric space. Please note that there are various kinds of sets, and every set may adopt multiple distance functions to determine the corresponding distance.