1. Bases and closures under infinite sums
- Author
-
Bruhn, Henning and Georgakopoulos, Agelos
- Subjects
- *
BASES (Linear topological spaces) , *CLOSURE spaces , *PATHS & cycles in graph theory , *GRAPH theory , *RING theory , *FINITE fields - Abstract
Abstract: Motivated by work of Diestel and Kühn on the cycle spaces of infinite graphs we study the ramifications of allowing infinite sums in a module . We show that every generating set in this setup contains a basis if the ground set M is countable, but not necessarily otherwise. Given a family , we determine when the infinite-sum span of is closed under infinite sums, i.e.when . We prove that this is the case if R is a field or a finite ring and each element of M lies in the support of only finitely many elements of . This is, in a sense, best possible. We finally relate closures under infinite sums to topological closures in the product space . [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF