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By M. Capobianco, J. B. Frechen, M. Krolik

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For U, V ∈ S, let U ∗ V be the collection of all subsets A of FINL ∪ FINLv such that {x ∈ FINL ∪ FINLv : {y ∈ FINL ∪ FINLv : x < y & x ∪ y ∈ A} ∈ V} ∈ U. Then as in the case of the Galvin-Glazer theorem, one has that ∗ is an associative operation on S and that U → U ∗ V is continuous for all V ∈ S. Thus (S, ∗) is a topological semigroup, so we can apply the theory of minimal idempotents. Let SL be the closed subsemigroup of S consisting of all cofinite ultrafilters that concentrate on FINL and let SLv be the two sided ideal of S consisting of all cofinite ultrafilters that concentrate on FINLv .

Then for every basic open set [s, M ], there is N ∈ [s, M ] such that [s, N ] is either included or is disjoint from O. Proof. We shall use the already established facts about the combinatorial forcing applied to the set X = O, and we shall use the forcing lemmas relativized to the basic set [s, M ] in place of [∅, N] = N[∞] . Choose N ∈ [s, M ] that O-decides all sets of the form s ∪ t, where t is a finite subset of N/s = {n ∈ N : n > s}. If N O-accepts s, then we are done so let us assume it rejects it.

Proof. 52 to the closure M, the alternative [s, N ] ⊆ M is impossible. ∞ Let M = k=0 Mk be a decomposition of M into nowhere dense sets. Let [s, M ] be a given basic open set. Relativizing the argument, we may assume that in fact s = ∅. Using the fact that the conclusion of the lemma is true for the nowhere dense sets Mk , we build a decreasing sequence M ⊇ M0 ⊇ ... ⊇ Mk ⊇ ... such that n0 = minM0 < ... < nk = minMk < ... and such that [s, Mk ] ∩ Mn = ∅ for all n ≤ k and s ⊆ {n0 , . . , nk−1 }.

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