Eilenberg-Mac Lane-Steenrod-Milnor axioms | Steenrod-Eilenberg axioms

A family of functors H_n(·) 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 (X, A), there is a natural long exact sequence ...->H_n(A)->H_n(X)->H_n(X, A)->H_(n - 1)(A)->..., where the map H_n(A)->H_n(X) is induced by the inclusion map A->X and H_n(X)->H_n(X, A) is induced by the inclusion map (X, ϕ)->(X, A). The map H_n(X, A)->H_(n - 1)(A) is called the boundary map.