MacLane's planarity criterion for locally finite graphs

MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner [Graphentheorie, Bibliographisches Institut, Mannheim, 1970], we show that the topological cycle space introduced recently by Diestel and Kühn allows a verbatim generalisation of MacLane's criterion to locally finite graphs. This then enables us to extend Kelmans’ planarity criterion as well.