EDB — 0N1

Exercises

1. [0N1] Prerequisites:[0PS]↺↻.Note:See also eserc. [192]↺↻. Suppose \(𝜑:[0,∞)→[0,∞)\) is monotonic weakly increasing and subadditive, i.e. \(𝜑(t)+𝜑(s)≥ 𝜑(t+s)\) for each \(t,s≥ 0\); and suppose that \(𝜑(x)=0\) if and only if \(x=0\).

   Then \(𝜑◦ d\) is again a distance. Examples: \(𝜑(t)=\sqrt t\), \(𝜑(t)=t/(1+t)\), \(𝜑(t)=\arctan (t)\), \(𝜑(t)=\min \{ t,1\} \).

   Moreover show that if \(𝜑\) is continuous in zero then the associated topology is the same.  ¹

   Solution 1

   [0N2]↺↻