On countably compact group topologies without non-trivial convergent sequences on Q(κ) for arbitrarily large κ and a selective ultrafilter

Pseudocompactness and ultrafilters

“…This suggests that a good candidate for a torsion-free group that admits a 𝑝−compact topology might be a divisible group, such as Q. Indeed, Bellini, Rodrigues and Tomita recently showed that, if 𝑝 is a selective ultrafilter and 𝜅 is a cardinal such that 𝜅 = 𝜅 𝜔 , then Q (𝜅) (the direct sum of 𝜅 copies of Q) admits a 𝑝−compact group topology without non-trivial convergent sequences [BRT21b]. Our first result in this regard is that divisibility can be dropped: Proposition 1.…”

“…That is, 𝐻 will be a 𝑝−compact subgroup of Q (c) , without non-trivial convergent sequences, which contains an element not divisible (in 𝐻 ) by any 𝑛 ∈ 𝜔. We will use a construction similar to the one made in [BRT21b].…”

“…• 𝐼 ⊂ 𝐽 0 be such that {[𝑓 𝜉 ] 𝑝 ∶ 𝜉 ∈ 𝐼 } ∪ {[𝜒 ⃗ 𝜇 ] 𝑝 ∶ 𝜇 ∈ c} is a Q−basis for ult 𝑝 (Q (c) ). The next result appears in [BRT21b]. The difference will be that we wish to control the value of the homomorphisms 𝜙 𝛽 , when and (𝑠 𝜇 ∶ 𝜇 ∈ c) of rational numbers, where all but finitely many are 0, such that Moreover, given 𝑛 ∈ 𝜔, suppose that there exists 𝜈 ∈ 𝐻 so that 𝜒 𝜔+𝜔 = 𝑛𝜈.…”

“…Now we may prove the following theorem. Its proof is similar to the proof of Theorem 3.7 in [BRT21b], but we use the changes made to the homomorphisms to prove that we can find a closed subgroup 𝐻 of Q (c) and an element in 𝐻 which is not divisible by any 𝑛 ∈ 𝜔.…”