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We extend to infinite graphs the matroidal characterization of finite graphduality, that two graphs are dual iff they have complementary spanning trees insome common edge set. The naive infinite analogue of this fails.  The key in an infinite setting is that dual trees must share between them notonly the edges of their host graphs but also their ends: the statement that aset of edges is acyclic and connects all the vertices in one of the graphs iffthe remaining edges do the same in its dual will hold only once each of the twographs' common ends has been assigned to one graph but not the other, and'cycle' and 'connected' are interpreted topologically in the space containingthe respective edges and precisely the ends thus assigned.  This property characterizes graph duality: if, conversely, the spanning treesof two infinite graphs are complementary in this end-sharing way, the graphsform a dual pair.

Dual trees must share their ends