We present a necessary condition for a pair of \(\mathcal{C}(K)\) spaces to be isomorphic in terms of topological properties of Cantor–Bendixon derivatives of K. This in particular gives completely new information about the perfect kernels of such K. In the process, we extend known lower estimates of the Banach–Mazur distance between a pair of spaces of continuous functions from the case of scattered compact spaces to a more general setting. Next, we apply this general result to deduce some new information about isomorphisms of spaces of continuous functions over Eberlein compacta of height ω + 1. Further, we show that isomorphisms of \(\mathcal{C}(K)\) spaces preserve the spread of K, and we also prove some new results for pairs of spaces of continuous functions whose Banach–Mazur distance is less than 3.
