The length-ω1 open game quantifier propagates scales

[T ] is just a typical closed set in the topology on ω1 whose basic neighborhoods are the sets of the form Np = {f : ω1 → ω | p ⊆ f}, where p : α → ω for some countable α. Associated to T is a closed game G(T ) on ω of length ω1: at round α in G(T ), player I plays mα, then player II plays nα. Letting f(α) = 〈mα, nα〉 for all α < ω1, we say that player II wins the run of G(T ) determined by f iff f ∈ [T ]. (We code sequences from ω by prime powers, so that 〈n0, ..., nk〉 = ∏ i≤k p ni+1 i , and the decoding is given by (n)i = (exponent of pi in n)-1, where pi is the i th prime.) Of course, not all such closed games are determined; indeed, every game of length ω can be regarded as a clopen game of length ω1. However, the determinacy of G(T ) for definable T does follow from a large cardinal hypothesis, in virtue of the following beautiful theorem of Itay Neeman: