Eilenberg-Steenrod Axioms

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).