  For a generalized Cantor set $E(\omega)$ with respect to a sequence
$\omega=\{ q_n \}_{n=1}^{\infty} \subset (0,1)$, we consider Riemann surface
$X_{E(\omega)}:=\hat{\mathbb{C}} \setminus E(\omega)$ and metrics on
Teichm\"uller space $T(X_{E(\omega)})$ of $X_{E(\omega)}$. If $E(\omega) =
\mathcal{C}$ ( the middle one-third Cantor set), we find that on
$T(X_{\mathcal{C}})$, Teichm\"uller metric $d_T$ defines the same topology as
that of the length spectrum metric $d_L$. Also, we can easily check that $d_T$
does not define the same topology as that of $d_L$ on $T(X_{E(\omega)})$ if
$\sup q_n =1$. On the other hand, it is not easy to judge whether the metrics
define the same topology or not if $\inf q_n =0$. In this paper, we show that
the two metrics define different topologies on $T(X_{E(\omega)})$ for some
$\omega=\{ q_n \}_{n=1}^{\infty}$ such that $\inf q_n =0$.
