Long Exact Sequence of a Pair Axiom

One of the Eilenberg-Steenrod Axioms. It states that, 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.

See also Eilenberg-Steenrod Axioms