In Hatcher's book on vector bundles and K-theory, page 55, in order to extend the external product to the relative form, he uses the following identification:
$X/A\wedge Y/B= (X\times Y)/(X\times B\cup A\times Y)$,
For $X,Y$ compact Hausdorff spaces with $A,B$ closed subspaces of $X$ and $Y$, respectively.
I guess the identification is done by mapping the pair $([x],[y])$ to $[(x,y)]$. But I don't know why it is enough, and this identification seems strange to me, since the smash product requires working with pointed spaces!
So, if I understand correctly, $X/A\wedge Y/B= \frac{X/A\times Y/B}{([x_0]\times Y/B) \ \cup \ (X/A\times [y_0]) }$ and I don't see the role of $(x_0,y_0)$ in $(X\times Y)/(X\times B\cup A\times Y)$.
Thank you very much for any help, this topic is new for me.
In fact, the identification involves pointed spaces, but it has nothing to do with special points $x_0 \in X, y_0 \in Y$. The space $X/A$ is regarded as a pointed space by taking as a basepoint the common equivalence class $[A]$ of all $a \in A$, similarly $Y/B$. The space $X/A \wedge Y/B$ has as a basepoint the common equivalence class $e = [\eta]$ of all $\eta \in X/A \times \{ [B] \} \cup \{ [A] \} \times Y/B$, the space $X \times Y / X \times B \cup A \times Y$ the common equivalence class $x = [\xi]$ of all $\xi \in X \times B \cup A \times Y$.
The map $p : X \times Y \to X/A \times Y/B \to X/A \wedge Y/B$ identifies $X \times B \cup A \times Y$ to $e$. It therefore induces a continuous bijection $p' : X \times Y / X \times B \cup A \times Y \to X/A \wedge Y/B$. The spaces $X \times Y / X \times B \cup A \times Y$ as well as $X/A, Y/B$ and $X/A \wedge Y/B$ = $(X/A \times Y/B) / (X/A \times \{ [B] \} \cup \{ [A] \} \times Y/B)$ are compact. Therefore $p'$ is a homeomorphism.