Carlsson in K-theory handbook, page 28.
Locally compact topological spaces (Carlsson)
Defined similarly to ordinary homology, but with the the defining chain complexes taken to be “locally finite”. The resulting homology groups are functorial with respect to proper continuous maps, and proper homotopy invariant. It is formally dual to Cohomology with compact supports. It agrees with ordinary homology on compact spaces.
Example: For , the only nontrivial group sits in degree .
Carlsson claims, without going into details, that one can adapt the classical use of spectra to interpret locally finite homology as a representable functor.
arXiv: Experimental full text search
AT (Algebraic topology)
nLab page on Locally finite homology