gn.general topology - Do Smash Products and Quotients Commute?

The easiest way I know to say what is going on is to resort to looking at
"products" of pairs:
$$
(X, A) times (Y, B) = ( Xtimes Y , Atimes Y cup Xtimes B).
$$
The point of this notation is that the functor $(X, A) mapsto (X/A, *)$
carries $(X, A) times (Y, B)$ to $X/A wedge Y/B$. We can iterate this procedure,
and I'll write $T^n(Y,X)$ for the subspace of $Y^n$
satisfying
$$
(Y, X)^n = ( Y^n, T^n(Y, X)).
$$
Thus $(Y/X)^{wedge n} = Y^n /T^n(Y,X)$.

You can easily check that
$$
T^n( Y, X) = lbrace (y_1, ldots, y_n) mid y_i in X mbox{for at least one $i$}rbrace.
$$

On the other hand $Y^{wedge n}/X^{wedge n}$ is the quotient
of $Y^n$ by the subspace
$$
T^n(Y,*) cup X^n,
$$
which is different (unless $X = *$).