Countable Products of Spaces of Finite Sets

σn(Γ) = {x ∈ {0, 1} Γ : |supp(x)| ≤ n}. Here supp(x) = {γ ∈ Γ : xγ 6= 0}. This is a closed, hence compact subset of {0, 1}, which is identified with the family of all subsets of Γ of cardinality at most n. In this work we will study the spaces which are countable products of spaces σn(Γ), mainly their topological classification as well as the classification of their Banach spaces of continuous functions.