A function f: X → Y is a quasi-homeomorphism if there exists a constant K > 0 such that for every ε > 0, there is a δ > 0 such that for all x,y in X with d_X(x,y) < δ, K^{-1}d_Y(f(x),f(y)) < d_X(x,y)^α < Kd_Y(f(x),f(y)).
A function f: X → Y is a quasi-homeomorphism if there exists a constant K > 0 such that for every ε > 0, there is a δ > 0 such that for all x,y in X with d_X(x,y) < δ, K^{-1}d_Y(f(x),f(y)) < d_X(x,y)^α < Kd_Y(f(x),f(y)).