We study the poset ℬ(G) of all precompact Hausdorff group topologies on an infinite group G and its subposet ℬσ(G) of topologies of weight σ, extending earlier results of Berhanu, Comfort, Reid, Remus, Ross, Dikranjan, and others. We show that if ℬσ(G) ≠ 0 and 2|G/G′| = 2|G| (in particular, if G is abelian) then the poset [2|G|]σ of all subsets of 2|G| of size σ can be embedded into ℬσ(G) (and vice versa). So the study of many features (depth, height, width, size of chains, etc.) of the poset ℬσ(G) is reduced to purely set-theoretical problems. We introduce a cardinal function Dede(σ) to measure the length of chains in [X]σ for |X|>σ generalizing the well-known cardinal function Ded(σ). We prove that Dede(σ) = Ded(σ) iff cf Ded(σ) ≠ σ+ and we use earlier results of Mitchell and Baumgartner to show that Dede(א1) = Ded(א1) is independent of Zermelo-Fraenkel set theory (ZFC). We apply this result to show that it cannot be established in ZFC whether ℬא1(Z) has chains of bigger size than those of the bounded chains. We prove that the poset ℋא0(G) of all Hausdorff metrizable group topologies on the group G = ⊕א0 Z2 has uncountable depth, hence cannot be embedded into ℬא0(G). This is to be contrasted with the fact that for every infinite abelian group G the poset ℋ(G) of all Hausdorff group topologies on G can be embedded into ℬ(G). We also prove that it is independent of ZFC whether the poset ℋא0(G) has the same height as the poset ℬא0(G). © 1998 Elsevier Science B.V.
Berarducci, A., Dikranjan, D., Forti, M., & Watson, S. (1998). Cardinal invariants and independence results in the poset of precompact group topologies. Journal of Pure and Applied Algebra, 126(1–3), 19–49. https://doi.org/10.1016/S0022-4049(96)00149-1