We introduce the Teichmüller space $T(E)$ of an ordered countable set $E$ of infinite number of distinct points on the Riemann sphere. We discuss the relation between the Teichmüller distance on $T(E)$ and a natural one on the configuration space for $E$. Also we give a system of global holomorphic coordinates for $T(E)$ when $E$ is determined from a finitely generated semigroup consisting of Möbius transformations with the totally disconnected forward limit set.