## 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 (1)

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 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).