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Let      ρ  \rho       be a metric on a space      X X       and let      s ≥  1 s\geq 1      . The function      ρ  s ( a , b ) = ρ  ( a , b ) s \rho ^s(a,b)=\rho (a,b)^s       is a quasimetric (it need not satisfy the triangle inequality). The function      inf ρ  s ( a , b ) \inf \rho ^s(a,b)       defined by the condition      inf ρ  s ( a , b ) = inf { ∑  0 n ρ  s ( z i , z i + 1 ) \inf \rho ^s(a,b)=\inf \{\sum _0^n \rho ^s(z_i,z_{i+1})            z 0 = a , z n = b } z_0=a, z_n=b\}       is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space      ( X , ρ  ) (X,\rho )      . We also give some examples showing how the topology of the space      ( X , inf ρ  s ) (X,\inf \rho ^s)       can change as      s s       changes.

Metrics 𝜌, quasimetrics 𝜌^{𝑠} and pseudometrics inf𝜌^{𝑠}