Long chains of topological group topologies—A continuation

We continue the work initiated in our earlier article (J. Pure Appl. Algebra 70 (1991) 53–72); as there, for G a group let B(G) (respectively N(G)) be the set of Hausdorff group topologies on G which are (respectively are not) totally bounded. In this abstract let A be the class of (discrete) maximally almost periodic groups G such that ¦G¦ = ¦GG′¦. We show (Theorem 3.3(A)) for G ϵ A with ¦G¦ = γ ⩾ ω that the condition that B(G) contains a chain C with ¦C¦ = β is equivalent to a natural and purely set-theoretic condition, namely that the partially ordered set 〈P(2γ), ⊆ 〉 contains a chain of length β. (Thus the algebraic structure of G is irrelevant.) Similar results hold for chains in B(G) of fixed local weight, and for chains in N(G). m 6.4. If T1 ϵ B(G) and the Weil completion 〈(G,T1〉 is connected, then for every Hausdorff group topology T0 ⊆ T1 with ω〈G,T0〉 < α1 = ω〈G,T1〉 there are 2α1-many gro topologies between T0 and T1. heorem 7.4. Let F be a compact, connected Lie group with trivial center. Then the product topology T0 on Fω is the only pseudocompact group topology on Fω, but there are chains C ⊆ B(Fω) and C′ ⊆ B(Fω) with ¦C¦ = (2c+ and ¦C′¦ = 2(c+)such that T0 ⊆ ∩C and T0 ⊆ ∩C′.