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).

• Clausen Function
• Clausen's Integral
• CLEAN Algorithm
• CLEAN Beam
• CLEAN Map
• Clebsch-Aronhold Notation
• Clebsch Diagonal Cubic
• Clebsch-Gordan Coefficient
• Clement Matrix
• Clenshaw Recurrence Formula
• Clifford Algebra
• Clifford's Circle Theorem
• Clifford's Curve Theorem
• Cliff Random Number Generator
• Clique
• Clique Number
• Clock Solitaire
• Closed Curve
• Closed Curve Problem
• Closed Disk
• Closed Form
• Closed Graph Theorem
• Closed Interval
• Closed Set
• Close Packing