Metric Space

Def 13.1

$(X,d)$ is called metric space
0) $X$ is set

1. $d:X\times X⟶[0,\alpha )$
   1. $d(x,y)=d(y,x)$
   2. $d(x,y)=0⟺x=y$
   3. $d(x,y)\le d(x,z)+d(z,y)$

3 standard choices of distance function 17.4

1. $d_{\infty }(x,y)={\max }_d|x(d)-y(d)|$
2. $d_2(x,y)=\sqrt{{\sum }_d|x(d)-y(d){|}^2}$
3. $d_1(x,y)={\sum }_{d=1}^d|x(d)-y(d)|$