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arXiv · 2606.21204

Borel complexity of isometry classes of $\mathcal{C}(K)$ spaces with countable compacta

Marek Cúth·Martin Doležal·Ondřej Kurka·Jakub Rondoš
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Abstract

For every countable compact space $K$, we determine the exact Borel complexity of the isometry class of the Banach space $\mathcal{C}(K)$. As a byproduct, we also determine the precise Borel complexity of the homeomorphism class of a fixed countable compact space $K$, improving earlier results of Cenzer and Mauldin. The above results provide concrete and natural examples of sets with arbitrarily high, still exactly determined, Borel complexity. Moreover, we find a new characterization of those real $L_1$-preduals that are isometric to $\mathcal{C}(K)$ for some zero-dimensional compact space $K$ and we determine the precise Borel complexity of $\mathcal{C}(2^{\mathbb{N}})$.

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Marek Cúth, Martin Doležal, Ondřej Kurka, Jakub Rondoš. 2026-06-19. Borel complexity of isometry classes of $\mathcal{C}(K)$ spaces with countable compacta. https://arxiv.org/abs/2606.21204

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