Published 2021-05-19
Keywords
- Dual groups,
- topological groups,
- reflections
How to Cite
Copyright (c) 2021 Revista Integración, temas de matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
In this paper we present a study of the duality of a group via reflections. We begin with the demonstration of a necessary condition for the continuity of the dual homomorphism of the homomorphism that goes from the group to its reflection, that is, if φ: G → ξ(G), it follows that φb: ξ[(G) → Gb is a continuous bijection for T ∈ ξ, where ξ is a reflective subcategory of the category of topological groups and ξ(G) is the reflection of G. Once the previous condition is met, it is shown that Gb ∼= ξ[(G), when G is either a compact group or a topological group Čech complete with φ: G → ξ(G) surjective and open or a locally compact topological group and φ: G → ξ(G) is surjective and open.
In the case of the dual reflections of metrizable topological groups, we rely on a result of Chasco [5] which implies that when G is a metrizable abelian topological group and H is a dense subgroup of G, then the dual groups Gb and Hb are topologically isomorphic.
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