Eilenberg–Steenrod axioms

Definition

A pair of spaces $(X, A)$ is a topological space $X$ with a subspace $A$. A map of pairs $f \colon (X, A) \to (Y, B)$ is a continuous map $f \colon X \to Y$ with $f(A) \subseteq B$. Write $X$ for the pair $(X, \emptyset)$. An (ordinary) homology theory consists of functors $h_n$, for $n \in \Z$, from pairs of spaces to abelian groups, and natural transformations $\partial \colon h_n(X, A) \to h_{n-1}(A)$. The Eilenberg–Steenrod axioms are the following conditions:

1. (Homotopy) If $f, g \colon (X, A) \to (Y, B)$ are homotopic through maps of pairs, then $f_* = g_*$.
2. (Exactness) For each pair, the sequence $\cdots \to h_n(A) \to h_n(X) \to h_n(X, A) \xrightarrow{\partial} h_{n-1}(A) \to \cdots$, with the unnamed maps induced by inclusions, is exact.
3. (Excision) If $Z \subseteq A$ and the closure of $Z$ lies in the interior of $A$, then the inclusion $(X \setminus Z, A \setminus Z) \to (X, A)$ induces isomorphisms $h_n(X \setminus Z, A \setminus Z) \to h_n(X, A)$.
4. (Dimension) For a one-point space $P$, $h_n(P) = 0$ for all $n \ne 0$.
5. (Additivity) If $X$ is the disjoint union of spaces $X_\alpha$, the inclusions induce an isomorphism $\bigoplus_\alpha h_n(X_\alpha) \to h_n(X)$.

The group $h_0(P)$ is the coefficient group of the theory.