Cardinal invariants and independence results in the poset of precompact group topologies Original Research Article

We study the poset image(G) of all precompact Hausdorff group topologies on an infinite group G and its subposet imageσ(G) of topologies of weight σ, extending earlier results of Berhanu, Comfort, Reid, Remus, Ross, Dikranjan, and others. We show that if imageσ(G) ≠ empty set︀ 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 imageσ(G) (and vice versa). So the study of many features (depth, height, width, size of chains, etc.) of the poset imageσ(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 image is independent of Zermelo-Fraenkel set theory (ZFC). We apply this result to show that it cannot be established in ZFC whether imageimage1(Z) has chains of bigger size than those of the bounded chains.