This must be an elementary question, as I couldn't find any proofs on the Internet, but I still can't do it. And yes, Hatcher says that the join is not actually associative for general topological spaces, but it is for locally Hausdorff ones. My definition of the join is a quotient of a product.

My problem is that I don't really understand the topology of the join. Is there a basis? What do open sets look like? I think I can find a bijection that is supposed to be a homeomorphism between $(A*B)*C$ and $(B*C)*A$ (this should suffice, as commutativity seems easy): $$[[a,b,t],c,u] -> [[b,c,\frac{u}{1-(1-t)(1-u)}],a,(1-t)(1-u)].$$

Function in the third coordinate is not continuous at (0,0), so we can't just define it in the product and then push it down to the quotient. At (0,0) we can define it to be anything(e.g. $\frac{1}{2}$) as the 1 in the last coordinate collapses the first space ($B*C$) anyway.

The context of this question: I am trying to prove that $S^p * S^q = S^{p+q+1}$ and my idea was first to prove associativity and then just expand everything as multiple joins of $S^0$. There may be easier ways to prove the fact about spheres, but after having been stuck on this I really want to see the proof of associativity. The question about spheres is from Rolfsen's Knots and Links.