Invariance of compactness for the Bohr topology

✍ Scribed by Salvador Hernández; Sergio Macario

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 125 KB
• Volume: 111
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00196-0

✦ Synopsis

We define the g-extension of a topological Abelian group G as the set of all characters on G such that the restriction to every equicontinuous subset of G is continuous with respect to the pointwise convergence topology. A g-group is a topological Abelian group (G, τ ) such that its g-extension coincides with its completion. The Bohr topology of a topological group (G, τ ) is the topology that the group inherits as a subset of its Bohr compactification. A topological group (G, τ ) respects a property P if the subsets A of G that satisfy the property P are exactly the same for the Bohr topology and for the original topology of the group [Trigos-Arrieta, J. Pure Appl. Algebra 70 (1991) 199]. All groups here are assumed to be Abelian. We prove that every complete g-group when endowed with its Bohr topology is a µ-space. As a consequence, we obtain that for a complete g-group the properties of respecting functionally boundedness, pseudocompactness, countable compactness and compactness are all equivalent and a characterization of this property is also provided. Finally, we extend a theorem of Rosenthal about the existence of sequences equivalent to the 1 -basis. We prove that for a Čech-complete g-group the property of respecting compactness is equivalent to the existence of conveniently placed sequences equivalent to the 1 -basis.