By Dress A.W.M., Koolen J.

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It asks if the equality p = t is provable in ZFC. Since t = t(Fr⊥ ) it certainly connected with the interplay between the character χ(F ) of a semiﬁlter F and the tower number t(F ⊥ ) of its dual. 1. If F is a semiﬁlter with χ(F ) < d, then t(F ⊥ ) ≥ ℵ1 . Proof. We should show that each decreasing sequence (An )n∈ω in F ⊥ has a pseudointersection in F ⊥ . For any f : ω → ω consider the pseudointersection An ∩ [0, f (n)] Af = n∈ω of the sequence (An ). We claim that Af ∈ F ⊥ for some f . Fix a subfamily B ⊂ F of size |B| = ℵ0 · χ(F ) < d such that each F ∈ F contains some B ∈ B.

2. 15. Let X be a non-discrete subspace of βω with a non-isolated point x ∈ X. Show that a) χ(x, X) > ℵ0 , b) χ(x, X) ≥ p; c) χ(x, X) ≥ min{r, d}. 10). Finally, we describe closed subsets of βω \ ω = UF with help of ﬁlters. We recall that the pseudocharacter ψ(A; X) of a subset A of a topological space X is the smallest size |U| of a family U of open subsets of X with A = ∩U. If ψ(A; X) < κ (resp. ψ(A; X) ≤ κ) for some cardinal κ, then we shall say that A is a G<κ -subset (resp. G≤κ -subset) of X.

By deﬁnition, • p, the pseudointersection number is equal to the smallest size |F | of a strongly centred family F ⊂ [ω]ω , no inﬁnite pseudointersection; • t, the tower number, is the smallest size of a tower, that is a ⊂∗ -decreasing transﬁnite sequence (Tα )α<κ of inﬁnite subsets of ω, having no inﬁnite pseudointersection. 8. Show that t is equal to the smallest size |F | of a linearly ordered subfamily F ⊂ ([ω]ω , ⊂∗ ) having no inﬁnite pseudointersection. 9. Prove the inequalities ω1 ≤ p ≤ t ≤ b ≤ d.

### 4n-10 by Dress A.W.M., Koolen J.

by Michael

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