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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 $M$.

Isomorphic classification of projective tensor products of spaces of continuous functions