On a Theorem of Van Mill

We fix an error in the proof of a theorem of Van Mill about homeomorphisms between compactifications of ω with small weight. Theorem 4.2. Let aω and bω be compactifications of ω. Assume that (1) there is a retraction r : aω → aω \ ω, (2) there is a retraction s : bω → bω \ ω, (3) f : aω \ ω → bω \ ω is a homeomorphism. If the weight of aω \ ω is less than p, then f can be extended to a homeomorphism f̄ : aω → bω. The above is Van Mill’s Theorem 4.2 from [5]. Andrea Medini noticed an error in Van Mill’s proof. In short, Claim 3 of that proof is wrong, though plausible at first reading. Upon close inspection, it is seen that while π[r[Uα,i] ∩M1] ⊆∗ s[Vα,i] ∩N1 and π[r[Uα,i] ∩M2] ⊇∗ s[Vα,i] ∩N2 are true, neither π[r[Uα,i] ∩ ω] ⊆∗ s[Vα,i] nor π[r[Uα,i] ∩ ω] ⊇∗ s[Vα,i] is true in general. I propose the following proof of the theorem. Lemma 1. Suppose A and B are boolean subalgebras of P(ω) such that both A and B has size less than p and the empty set is the only finite set in A ∪ B. If H is an isomorphism from A to B, then there is a permutation g of ω such that g[A] =∗ H(A) for all A ∈ A. Proof. Fix H as above. By Bell’s Theorem [1], it suffices to exhibit a σ-centered forcing P and a set D of fewer than p-many dense subsets of P such that a map g as above can be constructed from an arbitrary filter G of P that meets every set in D. Let P be a forcing order with conditions of the form p = 〈gp,Fp〉 where gp is an injective finite partial function from ω to ω and Fp is a finite subalgebra of A. Order P by declaring q ≤ p if gq ⊇ gp, Fq ⊇ Fp, and (gq \ gp)[U ] ⊆ H(U) for all U ∈ Fp. If p, q ∈ P and gp = gq, then 〈gp, E〉 is a common extension of p and q if E is a finite subalgebra of A such that E ⊇ Fp ∪ Fq. Thus, P is σ-centered. Moreover, we could have chosen E to contain an arbitrary element of A, so the set DA = {p ∈ P : A ∈ Fp} is dense for all A ∈ A. For each n < ω, let D′ n and D ′′ n respectively denote {p ∈ P : n ∈ dom(gp)} and {p ∈ P : n ∈ ran(gp)}. Given n < ω and p ∈ P \ D′ n, let A be the unique atomic