Combinatorics of open covers (VII): Groupability

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Selected results on selection principles (2012, background): “Selection Property Game Partition relation Source 1 S1(O,O) G1(O,O) Ω → (Λ) 2 m [58], [69] 2 S1(Ω,Ω) G1(Ω,Ω) Ω → (Ω) n m [70], [69] 3 S1(B,B) G1(B,B) BΩ → (B) 2 m [75] 4 S1(K,K) ? K → (K) 2 2 [20] 5 S1(Ω,Γ) G1(Ω,Γ) Ω → (Γ) 2 m [30], [69] 6 S1(K,Γ) G1(K,Γ) K → (Γ) n m [20], [17] 7 S1(K,Γk) G1(K,Γk) K”
Statistical convergence in topology (2008, methodology): “6 may be viewed as statistical versions of results from [32] (see [32, Theorems 21 and 26]) concerning characterizations of the Gerlits–Nagy and Hurewicz properties in all finite powers.”
MENGER'S AND HUREWICZ'S PROBLEMS: SOLUTIONS FROM "THE BOOK" AND REFINEMENTS (2009, unknown): “Let s, a ∈ [N]∞. s slaloms8 a if a∩ [s(n), s(n+1)) 6= ∅ for all but finitely many n. s slaloms a set Y ⊆ [N]∞ if it slaloms each a ∈ Y .”
Selection principles and hyperspace topologies (2005, unknown): “A countable open cover U of a space X is weakly groupable [17] (respectively k-weakly groupable [5]) if there is a partition U = {Un: n ∈ N} such that each Un is finite, Um ∩ Un = ∅ whenever m = n, and each finite (respectively compact) subset of X is covered by Un for some n ∈N.”
Scales, fields, and a problem of Hurewicz (2005, background): “In [22] it is proved that S1(,) is equivalent to (∀n, k) → ()k .”