The Tubby Torus as a Quotient Group

• Main Author: Sidney A. Morris
• Format: Article
• Language: English
• Published: MDPI AG 2020-01-01
• Series: Axioms
• Subjects: torus; tubby torus; separable quotient problem; locally convex space; nuclear space; banach space; pontryagin duality; weak topology
• Online Access: https://www.mdpi.com/2075-1680/9/1/11

Let <i>E</i> be any metrizable nuclear locally convex space and <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> the Pontryagin dual group of <i>E</i>. Then the topological group <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if <i>E</i> does not have the weak topology. This extends results in the literature related to the Banach&#8722;Mazur separable quotient problem.