In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent.
Here, we say that two Riemann surfaces are quasiconformally equivalent if there is a quasiconformal homeomorphism between them.
Hence, at the first stage of the theory, we have to know a condition for Riemann surfaces to be quasiconformally equivalent.
The condition is quite obvious if the Riemann surfaces are topologically finite.
On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather difficult.
We consider geometric conditions for the quasiconformal equivalence of open Riemann surfaces.
We also discuss the quasiconformal equivalence of regions which are complements of some Cantor sets, e. g., the limit sets of Schottky groups and the Julia sets of some rational functions.