---
tag: 4JG5Y1E5F3
kind: definition
name: Eilenberg–Steenrod axioms
msc: [55N40]
defines: YHK51GHK2T
uses: [082GHNPJ53, F2F143KQ2M, JB7ZBFBA9C, 0QEAJB91GZ, 0RZH6W98HK, M0Q5CFJ7ZH]
sources:
  - url: https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_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|082GHNPJ53]], 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|F2F143KQ2M]] 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|JB7ZBFBA9C]].
3. (Excision) If $Z \subseteq A$ and the [[closure|0QEAJB91GZ]] of $Z$ lies in the [[interior|0RZH6W98HK]] 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.

## Remarks

The additivity axiom was added later by Milnor; it is automatic for finite disjoint unions.
A *cohomology theory* is defined dually by contravariant functors $h^n$ and natural maps $\delta \colon h^n(A) \to h^{n+1}(X, A)$, with products in place of direct sums in the additivity axiom.

Example: [[singular homology|M0Q5CFJ7ZH]] with coefficients in an abelian group $G$ satisfies all five axioms, and its coefficient group is $H_0(P; G) = G$.

Non-example: complex topological K-theory, written as a cohomology theory $K^n$, satisfies the first three axioms and additivity for finite unions, but $K^{-2}(P) \cong \Z$, so it fails the dimension axiom.