Abstract
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 XF and XG 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 [5].
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