ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

W ⊆ f −1 {0}. By (3) for stage α, it suffices to show that Dα ∩ ↑U ⊆ Dα ∩ ↑W . Suppose
Z ∈ Dα ∩ ↑U. Then W/Mα ⊆ (f −1 {0})/Mα ⊆ U/Mα ⊆ Z/Mα. Since Z ∈ Dα ⊆ Mα
and Z is cozero, we have W ⊆ Z. Thus, Dα ∩ ↑U ⊆ Dα ∩ ↑W .
Corollary 3.4.15. Let X be a k-adic compactum and U be a family of subsets of X
such that for all U ∈ U there exists V ∈ U such that V ∩ X \ U = ∅. Then U is almost
ω op -like. Hence, πNt(X) = χKNt(X) = ω.
Proof. Proceed as in the proof of Theorem 3.3.3. Use the above theorem instead of
Theorem 3.3.2.
Theorem 3.4.16. Let X be a homogeneous k-adic compactum with base A. Then A
contains an ω op -like base of X.
Proof. By homogeneity and Lemma 3.4.9, we have πχ(p, X) = w(X) for all p ∈ X. By
Lemma 3.3.19, we may assume A consists only of cozero sets. Proceed as in the proof of
Lemma 3.3.18. Replace 2 λ with a k-metrizable compactum Y and replace B with the set
of cozero subsets of Y . For the proof of (2) for stage α + 1, we need a different argument
that, given H ∈ Eα and N ∈ Σα, the set Dα ∩ N ∩ ↑H is finite.
Choose U ∈ Uα such that H = Eα,U; choose V ∈ Uα such that V ⊆ U. Since π Y N
is open by Lemma 3.4.13, we have (h −1 V )/N ⊆ (f −1 {0})/N ⊆ (h −1 U)/N for some
f ∈ C(Y ) ∩ N. Since f ∈ N, we have h −1 V ⊆ f −1 {0}. Choose β < α such that f ∈ Mβ.
By elementarity, we may choose W0 ∈ Aβ such that h −1 W0 ⊆ f −1 {0}. Choose W1 ∈ Vβ
such that W 1 ⊆ W0; choose W2 ∈ Uβ such that W 2 ⊆ W1. By (2) for stage α, it suffices
to prove Dα ∩ N ∩ ↑Eα,U ⊆ ↑Eβ,W2. Suppose G ∈ Dα ∩ N ∩ ↑Eα,U. Then we have
(f −1 {0})/N ⊆ (h −1 U)/N ⊆ Eα,U/N ⊆ G/N.
92