Very Large Families of Dense Pseudocompact Groups in Compact Product-Like Groups
by
Gerald Itzkowitz
Queens College

A compact group G is said to be product-like if there is a family { H_\alpha : \alpha in A, |A| = w(G) } of compact metric groups and a continuous homomorphism of G onto H = \prod_{\alpha in A} H_\alpha. One should note that in this case the cardinal of H, denoted by |H|, satisfies |H| = |G|. The main result is that if G is a compact product-like nonmetrizable group then G contains exp(|G|) pairwise distinct dense pseudocompact subgroups. One should note that this means that such compact groups contain the largest possible number of dense pseudocompact subgroups since exp(|G|) is exactly the number of subsets of G. Since all compact nonmetrizable groups that are either Abelian or connected are product-like all such groups contain exp(|G|) dense pseudocompact subgroups. A brief survey of known results related to this theorem will be offered as time permits. Included in the survey is a closely related theorem of Comfort and van Mill which shows that under suitably restricted hypotheses on a nonmetrizable compact group G the word 'pseudocompact' may be replaced by '\omega-bounded'.

Date received: June 24, 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 # caah-43.