Products of protopological groups

Julie C Jones

doi:10.1155/s016117120100727x

Abstract

Montgomery and Zippin saied that a group is approximated by Lie groups if every neighborhood of the identity contains an invariant subgroup H such that G/H is topologically isomorphic to a Lie group. Bagley, Wu, and Yang gave a similar definition, which they called a pro‐Lie group. Covington extended this concept to a protopological group. Covington showed that protopological groups possess many of the characteristics of topological groups. In particular, Covington showed that in a special case, the product of protopological groups is a protopological group. In this note, we give a characterization theorem for protopological groups and use it to generalize her result about products to the category of all protopological groups.

Highlights

Montgomery and Zippin [5] saied that a group is approximated by Lie groups if every neighborhood of the identity contains an invariant subgroup H such that G/H is topologically isomorphic to a Lie group
Covington [3] extended this concept to topological groups. She defined a protopological group as a group G with a topology τ and a collection ᏺ of normal subgroups such that (1) for every neighborhood U of the identity, there exists N ∈ ᏺ such that N ⊆ U and (2) G/N with the quotient topology is a topological group for every N ∈ ᏺ
In [2], Covington defines a t-protopological group as a protopological group (G, τ) with the additional requirement that the natural map ηN : G → G/N is an open map for all N ∈ ᏺ. She shows that the product of t-protopological groups is a t-protopological group. Her proof uses ideas different than those used in the proof that a product of topological groups is a topological group, it uses the fact that ηN : G → G/N is an open map for all N ∈ ᏺ

Summary

Introduction

Montgomery and Zippin [5] saied that a group is approximated by Lie groups if every neighborhood of the identity contains an invariant subgroup H such that G/H is topologically isomorphic to a Lie group. She defined a protopological group as a group G with a topology τ and a collection ᏺ of normal subgroups such that (1) for every neighborhood U of the identity, there exists N ∈ ᏺ such that N ⊆ U and (2) G/N with the quotient topology is a topological group for every N ∈ ᏺ. Her proof uses ideas different than those used in the proof that a product of topological groups is a topological group, it uses the fact that ηN : G → G/N is an open map for all N ∈ ᏺ.

Results

Conclusion