Countable products of spaces of finite sets

Abstract

We consider the compact spaces σn(Γ) of subsets of Γ 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. For an infinite set Γ and a natural number n, we consider the space 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}Γ, 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. Let T be the set of all sequences (τn) ∞ n=1 with 0 ≤ τn ≤ ω. When τ runs over T , στ (Γ) = ∏∞ 1 σn(Γ) τn runs over all finite and countable products of spaces σk(Γ). For τ ∈ T we will call j(τ) to the supremum of all n with τn > 0 and i(τ) to the supremum of all n with τn = ω. If τn i(τ). (2) Suppose i(τ) = ω. In this case, if i(τ ′) = ω, then στ (Γ) is homeomorphic to στ ′(Γ). This is not a complete classification and leaves the following question open: 2000 Mathematics Subject Classification. 46B50, 46B26, 54B10, 54D30.