Confused by Hatchers proof of Corollary 2.24

Question

Corollary 2.24 says: If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A \cap B) \rightarrow (X,A)$ induces isomorphisms $H_{n}(B,A \cap B) \rightarrow H_{n}(X,A)$ for all $n$.

Hatcher then gives a very brief proof that I don't comprehend:

Since the $CW$ pairs are good, Proposition 2.22 allows us to pass to the quotient spaces $B/A \cap B$ and $X/A$ which are homeomorphic, assuming we are not in the trivial case $A \cap B = \emptyset$.

Can someone please elaborate on this proof? In particular, why are those two spaces homeomorphic and how exactly do we apply proposition 2.22 here?

Answer 1

The inclusion $B \to X$ induces a map $B/(A \cap B) \to X/A$, and its inverse is induced by the map $X \to B/(A \cap B)$ that is the identity on $B - A$ and that sends $A$ to $A \cap B$.

You may then apply the proposition which states that for a good pair $(X,A)$, $H_n(X,A) = \tilde H_n(X/A)$.