The smash product of pointed spaces is not associative. (see for example this question)
I am interested in the following special case of this: Let $X$ and $Y$ be two topological spaces. Is $(S^1\wedge X)\wedge(S^1\wedge Y)\cong S^2\wedge(X\wedge Y)$?
I am aware that this is true in a convenient category such as compactly generated spaces, however the "ugliness" of certain topological spaces leads me to believe that this is probably not true in general.
Does the situation improve if we assume stuff about the basepoint? For example, is the statement true if we replace $X$ and $Y$ by $X_+=X\amalg\{*\}$ and $Y_+=Y\amalg\{*\}$?
I started thinking about this because I was trying to prove the isomorphism $Th(\xi\times\zeta)\cong Th(\xi)\wedge Th(\zeta)$ (proposition 3.7 here) for Thom spaces of vector bundles, which has this as a special case as $Th(\underline{\mathbb{R}}^n)\cong S^n\wedge X_+$. The Books I found about this topic either prove this only for compact base or work in the category of compactly generated spaces.