Homology theory and the exactness axiom

In this article we give a description of a certain new homology theory on fairly wide categories of topological spaces, as well as a survey of papers concerning this construction. In contrast to the Aleksandrov–Иech homology theory this theory satisfies all the Eilenberg–Steenrod axioms, including exactness. In the end it turns out to be equivalent to the Steenrod homology theory and very close to the Borel–Moore homology theory, being isomorphic to it when the coefficient module is finitely generated (without this condition the Borel–Moore theory is not well-defined). We show that many results of the Borel–Moore theory take their most definitive and natural form in the homology theory under discussion. The methods of sheaf theory, which are used in the Borel–Moore homology theory, can be applied just as effectively in our case.