Rothberger's property in all finite powers

Abstract

A space X has the Rothberger property in all finite powers if, and only if, its collection of ω-covers has Ramseyan properties. For s ∈ [N] <ℵ 0 and for B ∈ [N] ℵ 0 use s < B to denote that s = ∅ or max(s) < min(B). For s < B define [s, B] = {s ∪ C ∈ [N] ℵ 0 : s < C ⊆ B}. The family {[s, B]: s ⊂ N finite and s < B ∈ [N] ℵ 0 } forms a basis for a topology on [N] ℵ 0 . This is the Ellentuck topology on [N] ℵ 0 and was introduced in [3].

Ramsey Theory and the Borel Conjecture

“…We now focus on a specific strengthening of Borel's covering property, namely the Rothberger covering property in all finite powers. Much of the material in this section is proven in prior papers, including [39]. For the convenience of the reader we provide a proof of the implications depicted in Figure 2.…”

Algebra, selections, and additive Ramsey theory

“…If S 1 (Ω(X), Ω(X)) holds, then for each cover U ∈ Ω(X) and each finite coloring of the set [U] 2 , there is in Ω(X) a cover V ⊆ U such that the graph [V] 2 is monochromatic. Scheepers proved a large number of results of this type, including ones jointly with Ljubiša Kočinac and others (e.g., [14,25,20]).…”