Question

Define a metric space .Give an example of a metric space

Answer 1

A metric space is an ordered pair ${\displaystyle (M,d)}$,  where ${\displaystyle M}$ is a set and ${\displaystyle d}$  is a metric on ${\displaystyle M}$, i.e., a function

${\displaystyle d\,\colon M\times M\to \mathbb {R} }$ such that for any ${\displaystyle x,y,z\in M}$ , the following holds:

1. ${\displaystyle d(x,y)=0\iff x=y}$ (identity of indiscernibles)
2. ${\displaystyle d(x,y)=d(y,x)}$ (symmetry)
3. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ (subadditivity or triangle inequality).

An example of a metric space is $(\mathbb R,d),$ where $d(x,y)=|x-y|.$ Indeed,

1. ${\displaystyle d(x,y)=0\iff |x-y|=0 \iff x-y=0\iff x=y}$
2. ${\displaystyle d(x,y)=|x-y|=|-(y-x)|=|y-x|=d(y,x)}$
3. ${\displaystyle d(x,z)=|x-z=|x-y+y-z|\leq|x-y|+|y-z|= d(x,y)+d(y,z)}$