We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the end-faithful spanning trees. We also generalize Eulers theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.