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We present a necessary condition for a pair of $\mathcal{C}(K)$ spaces to beisomorphic in terms of topological properties of Cantor-Bendixon derivatives of$K$. This in particular gives a completely new information about the perfectkernels of such $K$. In the process, we extend known lower estimates of theBanach-Mazur distance between a pair of spaces of continuous functions from thecase of scattered compact spaces to a more general setting. Next, we apply thisgeneral result to deduce some new information about isomorphisms of spaces ofcontinuous functions over Eberlein compacta of height $\omega+1$. Further, weshow 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 functionswhose Banach-Mazur distance is less than $3$.

On isomorphisms of $\mathcal{C}(K)$ spaces and cardinal invariants of derivatives of $K$