ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

To complete the proof of the claim, let us verify (2) for stage α + 1. By (1) for stage
α + 1, it suffices to prove Dα+1 ∩ ↑H is finite for all H ∈ Dα+1. By (3), if H ∈ Dα,
then Dα+1 ∩ ↑H = Dα ∩ ↑H, which is finite by (1) and (2) for stage α. Hence, we may
assume H ∈ Eα. Since Eα is ω op -like, it suffices to show that Dα ∩ ↑H is finite. Since
Dα ⊆ Σα, it suffices to show that Dα ∩ N ∩ ↑H is finite for all N ∈ Σα. Let N ∈ Σα.
By Lemma 3.3.1, there exists G ∈ B∩N such that G ⊇ H and B∩N ∩↑H = B∩N ∩↑G;
hence, Dα ∩ N ∩ ↑H = Dα ∩ N ∩ ↑G. Since G ⊇ H ∈ C, we have G ∈ C. By (2) for stage
α, the set Dα ∩ N ∩ ↑G is finite; hence, Dα ∩ N ∩ ↑H is finite.
Since U ⊆ A, it suffices to prove that U is an ω op -like base of X. Suppose p ∈ V ∈ A.
Then there exists α < κ such that V ∈ Aα. Hence, there exists U ∈ Uα such that
p/Fα ∈ U/Fα ⊆ V/Fα; hence, p ∈ U ⊆ V . Thus, U is a base of X.
Let us show that U is ω op -like. Suppose not. Then there exists α < κ and U0 ∈ Uα
such that there exist infinitely many V ∈ U such that U0 ⊆ V . Choose U1 ∈ Uα such
that U 1 ⊆ U0. Suppose β < κ and U0 ⊆ V ∈ Uβ. Then Eα,U1 ⊆ h −1 U0 ⊆ h −1 V ⊆ Eβ,V .
By (1) and (2), Dκ is ω op -like; hence, there are only finitely many possible values for
Eβ,V . Therefore, there exist 〈γn〉n