  It is known that homology and inverse limit functors do not commute. In the
paper we consider this very problem and find its application for various
homology theories. In particular, on the category of general topological
spaces, there are defined exact homology functors induced by different non-free
cochain complexes. Relation between them and other classical homology theories
are given. In addition, for the defined homology functors the tautness and the
continuous properties are obtained.
