On the equivalence of weak quasisymmetry and quasisymmetry on non-connected sets

This paper studies the equivalence of the quasisymmetric mappings on non-connected sets. We introduce a generalized form of weak quasisymmetry and prove that, on a uniformly perfect metric space, a generalized weakly quasisymmetric mapping is quasisymmetric. We further improve this result on Cantor-like sets satisfying the small gap condition.