Banach-Stone-like results for combinatorial Banach spaces

We show that under a certain topological assumption on two compact hereditary families $\F$ and $\G$ on some infinite cardinal $κ$, the corresponding combinatorial spaces $X_\F$ and $X_\G$ are isometric if and only if there is a permutation of $κ$ inducing a homeomorphism between $\F$ and $\G$. We also prove that two different regular families $\F$ and $\G$ on $ω$ cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.

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