Metrization criteria for compact groups in terms of their dense subgroups

Details Metrization criteria for compact groups in terms of their dense subgroups Metrization criteria for compact groups in terms of their dense subgroups

2011-02-24

arxiv.org/abs/1102.5077v2

Mathematics

General Topology

The exposition has substantially improved. Remarks 5.6 and 5.7 are new. Three references added

Scientific paper

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G_\delta-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its G_\delta-dense subgroups is metrizable, thereby resolving a question by Hernandez, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Dominguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building G_\delta-dense subgroups without uncountable compact subsets in compact groups of weight \omega_1 (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.

Affiliated with

Also associated with

No associations

Rating

Metrization criteria for compact groups in terms of their dense subgroups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Metrization criteria for compact groups in terms of their dense subgroups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Metrization criteria for compact groups in terms of their dense subgroups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-708809