Generalized cohomology theory

Definition

Consider pairs of spaces $(X, A)$ and maps of pairs, as in the Eilenberg–Steenrod axioms, and write $X$ for $(X, \emptyset)$. A generalized cohomology theory consists of contravariant functors $h^n$, for $n \in \Z$, from pairs of spaces to abelian groups, and natural transformations $\delta \colon h^n(A) \to h^{n+1}(X, A)$, such that:

1. (Homotopy) Homotopic maps of pairs induce the same homomorphisms.
2. (Exactness) For each pair, the sequence $\cdots \to h^n(X, A) \to h^n(X) \to h^n(A) \xrightarrow{\delta} h^{n+1}(X, A) \to \cdots$, 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 $h^n(X, A) \to h^n(X \setminus Z, A \setminus Z)$.
4. (Additivity) If $X$ is the disjoint union of spaces $X_\alpha$, the inclusions induce an isomorphism $h^n(X) \to \prod_\alpha h^n(X_\alpha)$.

No condition is put on the groups $h^n(P)$ of a one-point space $P$.