# Hatcher 2.1.15

Hatcher, Algebraic Topology, Chapter 2, Section 1

15. For an exact sequence $A\to B\to C\to D\to E$ show that $C=0$ if and only if the map $A\to B$ is surjective and $D\to E$ is injective. Hence, for a pair of spaces $(X,A)$, the inclusion $A\hookrightarrow X$ induces isomorphisms on all homology groups if and only if $H_n(X,A)=0$ for all $n$.
Proof.

$\square$