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).

CITE THIS AS:

Weisstein, Eric W. "Eilenberg-Steenrod Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Eilenberg-SteenrodAxioms.html