Associativty of smash product on compact spaces 
Let $Y,Z$ be compact or $X,Z$ locally compact. Then the canonical bijection 
  $$ (X \wedge Y) \wedge Z \rightarrow X \wedge (Y \wedge Z) $$ 
  is a homeomoprhism. 

I can prove the case when $X,Z$ are locally compact using exponential law. But I don't see how one approaches the case when $Y,Z$ compact. Hints? 
 A: It is clear that we have surjections
$$p_{X;Y,Z} : X \times Y \times Z \stackrel{id_X \times q_{Y,Z}}{\longrightarrow} X \times (Y \wedge Z) \stackrel{q_{X,Y \wedge Z}}{\longrightarrow} X \wedge (Y \wedge Z)$$
$$p_{X,Y,Z} : X \times Y \times Z \stackrel{q_{X,Y} \times id_Z}{\longrightarrow} (X \wedge Y) \times Z \stackrel{q_{X \wedge Y,Z}}{\longrightarrow} (X \wedge Y) \wedge Z$$
where the $q$-maps are quotient maps. Moreover, there is a bijection $\tau :  X \wedge (Y \wedge Z) \to  (X \wedge Y) \wedge Z$ such that $\tau \circ p_{X;Y,Z} = p_{X,Y;Z}$.
It therefore suffices to show that both $p$-maps are quotient maps. In turn it suffices to show that $id_X \times q_{Y,Z}$ and $q_{X,Y} \times id_Z$ are quotient maps.
If $X,Z$ are locally compact, this is well-known. It remains to show that if $Y,Z$ are compact, then $id_X \times q_{Y,Z}$ is a quotient map. In this case we have $q_{Y,Z} : Y \times Z \to Y \wedge Z = (Y \times Z)/(Y \vee Z)$ where $Y \vee Z$ is compact. Now apply exercise 16 in section 2.2 in the book "Algebraic Topology" by tom Dieck.
Frankly, I have not done the exercise, but it seems to be straightforward.
See When is the product of two quotient maps a quotient map?
Noting that closed maps are quotient maps, see also Ronnie Brown's answer to https://mathoverflow.net/q/93679.
