Cardinal Numbers Associated With Dense Pseudocompact Groups
by
Gerald L. Itzkowitz
Queens College/C.U.N.Y.

A brief survey will be offered on the progress to date in the solution of two interrelated problems concerning dense proper pseudocompact (countably compact, \omega-bounded) subgroups of compact nonmetrizable groups. These are: Does every nonmetrizable compact group contain such a subgroup and if a compact group has such a subgroup how large may a distinguished family of such subgroups be? We will then present some theorems that may be deduced from these results. One such theorem is that each compact nonmetrizable group G that is connected or Abelian contains a family of 2^|G| distinct dense pseudocompact subgroups. Lower bounds are also given for the number of distinct countably compact subgroups in all groups considered in the first section, where it is shown that each such nonmetrizable compact group contains at least 2^\omega₁ distinct dense countably compact subgroups. Finally we will give some examples and pose a number of questions.

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-42.