Long Chains of Topological Group Topologies-A Continuation
by
W. W. Comfort
Wesleyan University
Coauthors: Dieter Remus

We continue work initiated in our earlier article [J. Pure and Applied Algebra 70 (1991), 53-72]. For G a group let B(G) [resp., N(G)] be the set of Hausdorff group topologies on G which are [resp., are not] totally bounded, and let A be the class of (discrete) maximally almost periodic groups G such that |G| = |G/G'|.

Theorem. For G in A with |G| = \gamma >= \omega, the condition that B(G) contains a chain C with |C| = \beta is equivalent to a natural and purely set-theoretic condition, namely that the partially ordered set <P(2^\gamma), subset or equal > contains a chain of length \beta. (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. If T₁ in B(G) and the Weil completion

<G, T1 >

is connected, then for every Hausdorff group topology T₀ subset or equal T₁ with w <G, T₀ > < \alpha₁ = w <G, T₁ > there are 2^\alpha₁-many group topologies between T₀ and T₁.

Theorem. Let F be a compact, connected Lie group with trivial center. Then the product topology T₀ on F^\omega is the only pseudocompact group topology on F^\omega, but there are chains C subset or equal B(F^\omega) and C' subset or equal B(F^\omega) with |C| = (2^c)⁺ and |C'| = 2^(c+) such that T₀ subset or equal \cap C and T₀ subset or equal \cap C'.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-16.