A family of functors 
from the category of
pairs of topological spaces and continuous maps,
to the category of Abelian
groups and group homomorphisms satisfies the Eilenberg-Steenrod axioms if the
following conditions hold.
1. long exact sequence of a pair axiom. For every pair 
, there is a natural long exact sequence
 |
where the map 
is induced by the inclusion
map 
and 
is induced by the inclusion map 
. The map 
is called the boundary
map.
2. homotopy axiom. If 
is homotopic to 
, then their induced
maps 
and 
are the same.
3. excision axiom. If 
is a space with subspaces 
and 
such that the set closure of 
is contained in the interior of 
, then the inclusion map 
induces an isomorphism 
.
4. dimension axiom. Let 
be a single point space. 
unless 
, in which case 
where 
are some groups. The 
are called the coefficients
of the homology theory 
.
These are the axioms for a generalized homology theory. For a cohomology theory, instead of requiring that 
be a functor, it is required
to be a co-functor (meaning the induced map points
in the opposite direction). With that modification, the axioms are essentially the
same (except that all the induced maps point backwards).
