Def 13.1

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

  1. 𝑑:𝑋×𝑋[0,𝛼)
    1. 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)
    2. 𝑑(𝑥,𝑦)=0𝑥=𝑦
    3. 𝑑(𝑥,𝑦)𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)

3 standard choices of distance function 17.4

  1. 𝑑(𝑥,𝑦)=max𝑑|𝑥(𝑑)𝑦(𝑑)|
  2. 𝑑2(𝑥,𝑦)=𝑑|𝑥(𝑑)𝑦(𝑑)|2
  3. 𝑑1(𝑥,𝑦)=𝑑𝑑=1|𝑥(𝑑)𝑦(𝑑)|