The join of $A$ and $B$ is the pushout of the diagram $$ CA \times B \gets A\times B \to A\times CB, $$ which can be formulated in either the pointed or unpointed topological category. This pushout is naturally a subspace of $CA\times CB$.
In the unpointed context, there is a homeomorphism $C(A * B) \cong CA \times CB$
extending the inclusion of the join into the product.
I have also seen it asserted in the pointed
context, with less convincing arguments.
In the pointed context, the case in which one
of the spaces is the one-point space $P$ seems to raise problems:
since $CP = P$ and $A*P = CA$,
$$
C(A * P) = C(CA)
\qquad
\mbox{while}
\qquad
CA \times CP \cong CA.
$$
So I am reluctantly inclined to believe that the pointed version is false.
Question: Is there a statement about pointed cones, products and joins that does roughly the same job as the unpointed homeomorphism $C(A* B) \cong CA\times CB$?
EDIT: Perhaps a solution is to restrict to CW complexes (or cell complexes), and prove that the cone and the reduced cone of such a space are homeomorphic, except for the one-point space (it's true for spheres, and probably induction will work).
