Countably compact group topologies on non-torsion Abelian groups of size continuum with non-trivial convergent sequences

Countably compact group topologies on arbitrarily large free Abelian groups

“…Given an arc U such that δ(U ) ≤ 1 2 , it follows that diam δ U = δ(U ). Our free Abelian groups will all be represented as directs sums of copies of the group of integers Z; we fix some notation.…”

“…Lemma 4.2. Assume we have a non-zero element d of G, an injective sequence r in G, and a countably infinite subset D of κ such that (1)…”

Countably compact group topologies on arbitrarily large free Abelian groups

“…Definition 5.2. Given an integer stack S and a natural number N , the N th root of S, written 1 N S, is obtained by keeping all the structure in S with the exception of the functions; these are divided by N . Thus a function f i,j,k ∈ S is replaced by 1 N f i,j,k in 1 N S for each 0 ≤ i < s, 0 ≤ j < r i and 0 ≤ k < r i,j and a function g l ∈ S is replaced by 1 N g l in 1 N S for each 0 ≤ l < t. A stack (unqualified) is then defined to be the N th root of an integer stack for some positive integer N .…”

“…By (#) in Lemma 5.3, each v j is an integer combination of the u i 's, therefore the inverse matrix of 1 N M, which we denote by N , has integer entries. Let ǫ ′ = ǫ • ( i,j<l |M i,j |) −1 and γ < ǫ ′ /(3M N s).…”

Countably compact group topologies on torsion-free Abelian groups

“…In fact, even in the construction of a countably compact group topology with nontrivial convergent sequences in nontorsion groups, a modification of the technique to produce countably compact groups without nontrivial convergent sequences; see Matheus Koveroff Bellini et al, 2019 andA. C. Boero, Castro-Pereira, et al, 2019. The first examples of countably compact groups without nontrivial convergent sequences were obtained by Hajnal and Juhász (Hajnal and Juhász, 1976) under CH.…”