Metrics ρ, quasimetrics ρ and pseudometrics inf ρs

Let ρ be a metric on a space X and let s≥1. The function ρ^s(a, b) = ρ(a, b)^s is a quasimetric (it need not satisfy the triangle inequality). The function inf ρ^ss(a, b) defined by the condition inf ρ^s(a, b) = inf(σⁿ ₀ρ^s(z_i, z_i+1) z₀ = 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). We also give some examples showing how the topology of the space (X, infρ^s) can change as s changes.