  We interpolate a new family of Teichm\"uller spaces $T_{\sharp}^X$ between
the universal Teichm\"uller space $T$ and its little subspace $T_0$, which we
call the Teichm\"uller space of piecewise symmetric homeomorphisms. This is
defined by prescribing a subset $X$ of the unit circle. The inclusion relation
of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of
$T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we
discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the
quasiconformal theory of Teichm\"uller spaces. We also consider the quotient
space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichm\"uller
space.
