The adaption of combinatorial duality to infinite graphs has
by the fact that while cuts (or cocycles) can be infinite, cycles are finite.
We show that these obstructions fall away when duality is reinterpreted on
the basis of a `singular' approach to graph homology, whose cycles are
defined topologically in a space formed by the graph together with its ends
and can be infinite. Our approach enables us to complete Thomassen's results
about `finitary' duality for infinite graphs to full duality, including his
extensions of Whitney's theorem.