Is the smash product also defined for an infinite family of pointed spaces? In section 7.9 of his book Algebraic Topology, tom Dieck defines this in the compactly generated context:
Tom Dieck defines a $k$-space to be a space $X$ such that every $k$-closed subset is closed in $X$ where $A\subset X$ is said to be $k$-closed if for every compact Hausdorff space $K$ and every continuous map $f: K\to X$, the preimage $f^{-1}(A)$ is closed in $K$.
For those $k$-spaces, the smash product is associative, but it is not associative in the category of all pointed spaces. However, one could still define
$$ \bigwedge_j X_j := \Big(\prod_j X_j\Big) \Big/ W_JX_j. $$
Is this a common definition? Or shouldn't we do this because of the missing associativity?
Thank you for any help and comments!