Eilenberg-Steenrod Axioms

A family of Functors from the Category of pairs of TopologicalSpaces and continuous maps, to the Category of Abelian Groups and grouphomomorphisms 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 bythe Inclusion Map . The Map is called the Boundary Map.

2. Homotopy Axiom. If is homotopic to , then their InducedMaps and are the same.

3. Excision Axiom. If is a Space with Subspaces and such that theClosure of is contained in the interior of , then the Inclusion Map inducesan 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 aFunctor, it is required to be a co-functor (meaning the Induced Map points in the opposite direction). Withthat modification, the axioms are essentially the same (except that all the induced maps point backwards).

• Gnomon
• Gnomonic Projection
• Gnomon Magic Square
• G-Number
• Go
• Goblet Illusion
• Goldbach Conjecture
• Golden Mean
• Golden Ratio
• Golden Ratio Conjugate
• Golden Rectangle
• Golden Rule
• Golden Section
• Golden Theorem
• Golden Triangle
• Goldschmidt Solution
• Golomb Constant
• Golomb-Dickman Constant
• Golomb Ruler
• Golygon
• Gomory's Theorem
• Gompertz Constant
• Gompertz Curve
• Gonal Number
• Goodman's Formula