  We propose a homology theory for locally compact spaces with ends in which
the ends play a special role. The approach is motivated by results for graphs
with ends, where it has been highly successful. But it was unclear how the
original graph-theoretical definition could be captured in the usual language
for homology theories, so as to make it applicable to more general spaces. In
this paper we provide such a general topological framework: we define a
homology theory which satisfies the usual axioms, but which maintains the
special role for ends that has made this homology work so well for graphs.
