Doing some exercises i often encounter arguments like $$H_0(S^2,A)=H_0(S^2/A, A/A).$$ Here $A$ is finite collection of points and $S^2$ is the two-dimensional sphere.
Why does this hold?
It seems, that this is a general fact, so I would be interested in a answer more applicable to more general cases than the provided example.
For a subspace $A\subseteq X$, if the inclusion $i:A\to{X}$ is a cofibration, the projection $$p:(X,A)\to (X/A,A/A)$$ induces an isomorphism $$p_*:H_n(X,A)\to H_n(X/A,A/A) \cong \tilde{H}_n(X/A)$$ on homology (and cohomology) for all $n$. Important special cases of this include $A$ being a subcomplex of a CW-complex $X$, which is your case, since we can realize a finite set of points $A\subseteq S^2$ as a subcomplex of $S^2$.
A proof of this can be found in Hatcher, page 124.