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

Abstract

We study the poset B (G) of all precompact Hausdorff group topologies on an infinite group G and its subposet B σ(G) of topologies of weight σ, extending earlier results of Berhanu, Comfort, Reid, Remus, Ross, Dikranjan, and others. We show that if B σ(G) ≠ ∅ 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 B σ(G) (and vice versa). So the study of many features (depth, height, width, size of chains, etc.) of the poset B σ(G) is reduced to purely set-theoretical problems. We introduce a cardinal function Ded e ( σ) to measure the length of chains in [X] σ for ¦X¦> σ generalizing the well-known cardinal function Ded( σ). We prove that Ded e ( σ) = Ded( σ) iff cf Ded( σ) ≠ σ + and we use earlier results of Mitchell and Baumgartner to show that Ded e( N 1) = Ded( N 1) is independent of Zermelo-Fraenkel set theory (ZFC). We apply this result to show that it cannot be established in ZFC whether B N 1 (Z) has chains of bigger size than those of the bounded chains. We prove that the poset H N 0 (G) of all Hausdorff metrizable group topologies on the group G = ⊕ N 0 Z 2 has uncountable depth, hence cannot be embedded into B N 0 (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 H N 0 (G) has the same height as the poset B N 0 (G).