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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]

Abstract: Let be a graph each edge of which is given a length . This naturally induces a distance between any two vertices , and we let  denote the completion of the corresponding metric space. It turns out that several well-studied topologies on infinite graphs are special cases of . Moreover, it seems that is the right setting for studying various problems. The aim of this paper is to introduce , providing basic facts, motivating examples and open problems, and indicate possible applications. Parts of this work suggest interactions between graph theory and other fields, including algebraic topology and geometric group theory. [Copyright &y& Elsevier]

Abstract: We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are “orthogonal” to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff''s first law also for ends. [Copyright &y& Elsevier]