Isomorphic classification of projective tensor products of spaces of continuous functions

Abstract: 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 $M$.