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
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).