  We prove that for infinite, countable, compact, Hausdorff spaces $K,L$,
$C(K)\widehat{\otimes}_\pi C(L)$ is isomorphic to exactly one of the spaces
$C(\omega^{\omega^\xi})\widehat{\otimes}_\pi C(\omega^{\omega^\zeta})$,
$0\leqslant \zeta\leqslant \xi<\omega_1$. We also prove that
$C(K)\widehat{\otimes}_\pi C(L)$ is not isomorphic to $C(M)$ for any compact,
Hausdorff space $M$.
