The coproduct of totally convex spaces (Q1089594)

The functor \({\hat 0}\) from the category \(Ban_ 1\) to the category TC of totally convex spaces has a left adjoint S. It is shown that S in turn has a left adjoint. This fact allows the description of the coproduct in a very special case and leads to the definition of the direct sum of objects in TC. The direct sum turns out to be the coproduct for separated totally convex spaces. In addition, it gives an explicit description of the interior of the coproduct in the general case. The description of the boundary of the coproduct is given by an explicit construction. Additionally, the authors investigate which of the interesting subcategories of TC are closed under coproducts and which of the results on coproducts remain valid for the category of finitely totally convex spaces.