Long chains of topological group topologies—A continuation

Abstract

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¦ = ¦ G G′ ¦ . 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 (2 γ), ⊆ 〉 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). Theorem 6.4. If T 1 ϵ B (G) and the Weil completion 〈(G,T 1〉 is connected, then for every Hausdorff group topology T 0 ⊆ T 1 with ω〈G, T 0〉 < α 1 = ω〈G, T 1〉 there are 2 α1-many gro topologies between T 0 and T 1. From Theorem 7.4. Let F be a compact, connected Lie group with trivial center. Then the product topology T 0 on F ω is the only pseudocompact group topology on F ω , but there are chains C ⊆ B (F ω) and C ′ ⊆ B (F ω) with ¦ C ¦ = (2 c+ and ¦ C ′¦ = 2 ( c+ ) such that T 0 ⊆ ∩ C and T 0 ⊆ ∩ C ′.