I apologize in advance if this question is too technical. I haven't found a reference in the literature yet, and it seems difficult enough that perhaps it has not been answered.


Let $A$, $B$, and $C$ be based topological spaces, all compactly generated and weak Hausdorff. Let $\wedge$ and $F$ denote smash product and space of based functions, respectively, modified in the usual way so as to produce CGWH spaces. Consider the "assembly" map
$$ A \wedge F(B,C) \rightarrow F(B, A \wedge C) $$
It is the adjoint of the smash product of the identity of $A$ and the evaluation map of $F(B,C)$.


Is this map always a closed inclusion?


(It is not hard to show that it is injective, has closed image, and that if $K$ is a closed subset of the domain, then $K \cup \{\*\}$ is a closed subset of the codomain. If you take $A$, $B$, and $C$ to be $[0,\infty)$, you appear to get a counterexample.)


EDIT: I believed earlier that this counterexample could maybe be fixed by making everything compactly generated, but that doesn't help, you can still produce a non-convergent sequence in the source that converges in the target to the basepoint, and this still produces a closed set whose image is not compactly closed. So this is actually not true! Juan, thanks for pointing me to Lewis's thesis.



is this map a closed inclusion?
Isn't this addressed in Lewis' Appendix A (on compactly generated spaces)? His Proposition 8.5 gives a criteria (for A compact), to get a closed inclusion. But your comment on adding the disjoint basepoint is important ("The adjunction od the disjoing basepoint is essential to the [...] result"). See his CounterExample 8.6 following the proposition.


