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
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
- 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).
See also Aleksandrov-Cech Cohomology
© 1996-9 Eric W. Weisstein