  On the category of compact metric spaces an exact homology theory was defined
and its relation to the Vietoris homology theory was studied by N. Steenrod
[S]. In particular, the homomorphism from the Steenrod homology groups to the
Vietoris homology groups was defined and it was shown that the kernel of the
given homomorphism are homological groups, which was called weak homology
groups [S], [E]. The Steenrod homology theory on the category of compact metric
pairs was axiomatically described by J.Milnor. In [Mil] the uniqueness theorem
is proved using the Eilenberg-Steenrod axioms and as well as relative
homeomorphism and clusres axioms. J. Milnor constructed the homology theory on
the category $Top^2_C$ of compact Hausdorff pairs and proved that on the given
category it satisfies nine axioms - the Eilenberg-Steenrod, relative
homeomorphis and cluster axioms (see theorem 5 in [Mil]). Besides, using the
construction of weak homology theory, J.Milnor proved that constructed homology
theory satisfies partial continuity property on the subcategory $Top^2_{CM}$
(see theorem 4 in [Mil]) and the universal coefficient formula on the category
$Top^2_C$ (see Lemma 5 in [Mil]). On the category of compact Hausdorff pairs,
different axiomatic systems were proposed by N. Berikashvili [B1], [B2],
H.Inasaridze and L. Mdzinarishvili [IM], L. Mdzinarishvili [M] and H.Inasaridze
[I], but there was not studied any connection between them. The paper studies
this very problem. In particular, in the paper it is proved that any homology
theory in Inasaridze sense is the homology theory in the Berikashvili sense,
which itself is the homology theory in the Mdzinarishvili sense. On the other
hand, it is shown that if a homology theory in the Mdzinarishvili sense is
exact functor of the second argument, then it is the homology in the Inasaridze
sense.
