Essential Density and Total Density in Topological Groups
by
W. W. Comfort
Wesleyan University, Middletown, Connecticut
Coauthors: Dikran Dikranjan (Università di Udine, Italia)

A subgroup H of a topological group G is said to be essential [resp. totally dense] in G if |H \cap N| > 1 [resp. H \cap N is dense in N] for every closed, non-trivial normal subgroup N of G. The essential density ED(G) [resp. total density TD(G)] of G is defined as the minimal cardinality of a dense essential [resp., totally dense] subgroup of G.

It has been shown recently by E. Boschi and D. Dikranjan that ED(G) = TD(G) for many topological groups G, including compact connected groups, LCA groups and totally minimal abelian groups. Here we answer questions suggested by these authors, showing that both ED and TD can rise upon passage to dense subgroups, and that the relation ED(G) = TD(G) can fail for suitable G. Specifically we show in ZFC:

Theorem 1. There is a compact, totally disconnected Abelian group K with a dense subgroup G such that TD(G)=ED(G)=|G| > TD(K)=ED(K).

Theorem 2. There is a compact, totally disconnected Abelian group K and a dense essential subgroup G such that ED(G) < TD(G).

In certain models of ZFC such pairs (K, G) may be chosen with G pseudocompact.

We do not know whether ED(G) = TD(G) holds for all compact groups, or for all countably compact Abelian groups.

Date received: May 23, 1999

Copyright © 1999 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 # cacl-19.