Convergent nets in abelian topological groups

Robert Ledet

doi:10.1155/s016117120100744x

Abstract

A net in an abelian group is called a T‐net if there exists a Hausdorff group topology in which the net converges to 0. This paper describes a fundamental system for the finest group topology in which the net converges to 0. The paper uses this description to develop conditions which insure there exists a Hausdorff group topology in which a particular subgroup is dense in a group. Examples given include showing that there are Hausdorff group topologies on ℝn in which any particular axis may be dense and Hausdorff group topologies on the torus in which S1 is dense.

Highlights

Let G be an abelian group and xα α∈A a net in G
Converges to 0? In the terminology of [1], we are placing the topology of a nonconstant net on the subspace ⊂ G and finding the associated Graev topology
We adopt the terminology of Zelenyuk and Protasov in the following

Summary

Introduction

Let G be an abelian group and xα α∈A a net in G. If 0 is the identity element in G, we can ask what is the finest group topology on G such that xα α∈A converges to 0? In the terminology of [1], we are placing the topology of a nonconstant net on the subspace ( xα α∈A ∪{0}) ⊂ G and finding the associated Graev topology. Ledet and Clark [2] developed a fundamental system approach to defining group topologies in which a sequence an

When there exists a Hausdorff group topology in which an

For each i

Then an a

Then if

Let αn

But for every subset