Eilenberg-Steenrod Axioms

A family of Functors $H_n(\cdot)$ 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

$$\ldots\to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \to \ldots,$$ (1)

where the Map $H_n(A) \to H_n(X)$ is induced by the Inclusion Map $A \to X$ and $H_n(X) \to H_n(X,A)$ is induced by the Inclusion Map $(X,\phi) \to (X,A)$. The Map $H_n(X,A) \to H_{n-1}(A)$ is called the Boundary Map.

2. Homotopy Axiom. If $f:(X,A) \to (Y,B)$ is homotopic to $g:(X,A) \to (Y,B)$, then their Induced Maps $f_*:H_n(X,A)\to H_n(Y,B)$ and $g_*:H_n(X,A)\to H_n(Y,B)$ are the same.

3. Excision Axiom. If $X$ is a Space with Subspaces $A$ and $U$ such that the Closure of $A$ is contained in the interior of $U$, then the Inclusion Map $(X \ U, A \ U) \to (X,A)$ induces an isomorphism $H_n(X \ U, A \ U) \to H_n(X,A)$.

4. Dimension Axiom. Let $X$ be a single point space. $H_n(X)=0$ unless $n=0$, in which case $H_0(X)=G$ where $G$ are some Groups. The $H_0$ are called the Coefficients of the Homology theory $H(\cdot)$.

These are the axioms for a generalized homology theory. For a cohomology theory, instead of requiring that $H(\cdot)$ 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