Countable products of spaces of finite sets

We consider the compact spaces sigma_n(I) of subsets of an uncountable set I of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.

💡 Research Summary

The paper studies the compact spaces σₙ(I), where I is an uncountable set of cardinality κ (κ ≥ ℵ₁) and σₙ(I) consists of all subsets of I of size at most n. Each σₙ(I) is zero‑dimensional, scattered of height n + 1, and therefore its space of real‑valued continuous functions C(σₙ(I)) is a Banach space with a very concrete structure. The authors first give a precise Banach‑space decomposition for a single σₙ(I): they prove that
\