  In the theory of Teichm\"uller space of Riemann surfaces, we consider the set
of Riemann surfaces which are quasiconformally equivalent.
  For topologically finite Riemann surfaces, it is quite easy to examine if
they are quasiconformally equivalent or not.
  On the other hand, for Riemann surfaces of topologically infinite type, the
situation is rather complicated.
  In this paper, after constructing an example which shows the complexity of
the problem, we give some geometric conditions for Riemann surfaces to be
quasiconformally equivalent.
  Our argument enables us to obtain the universal Schottky space which contains
all Schottky spaces, the deformation spaces of Schottky groups as the universal
Teichm\"uller space contains all Teichm\"uller spaces.
