Dense Subgroups of Prescribed Density

TL;DR

This paper proves that in any infinite compact group, for every cardinal between the group's density and weight, there exists a dense subgroup with that specific density, expanding understanding of subgroup structures.

Contribution

It establishes the existence of dense subgroups of prescribed density in infinite compact groups, a novel result in topological group theory.

Findings

Dense subgroups of any specified density exist in infinite compact groups.

The result applies to all cardinals between the group's density and weight.

This advances the understanding of subgroup structures in topological groups.

Abstract

Let $G$ be an infinite compact group. We prove that for every cardinal $\kappa$ between the density and the weight of $G$, there exists a dense subgroup of $G$ of density $\kappa$.