How Many \Omega-bounded Subgroups?
by
W.W. Comfort
Wesleyan University
Coauthors: Jan van Mill

A topological space is said to be \omega-bounded if each of its countable subsets has compact closure. It has been shown recently by Itzkowitz and Shakhmatov that for every compact Abelian group G of uncountable weight, and for every compact connected group G of uncountable weight, the set \Omega(G) of dense \omega-bounded subgroups of G satisfies |\Omega(G)| >= |G|. These authors asked whether their estimate |\Omega(G)| >= |G| may be improved to |\Omega(G)| = 2|G| for some or all such G. In the present paper we answer this question affirmatively for all compact groups G which are either Abelian or connected and which satisfy in addition the condition w(G) = (w(G))^\omega. We show also that every compact group G with w(G) >= log((2^c)⁺) satisfies |\Omega(G)| > 2^c.

Date received: December 30, 1995

Copyright © 1995 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 # caad-04.