To compute homotopy groups of one-point unions of based spaces (for instance, spheres) it is relevant to consider quotient spaces $\prod_{i=1}^{n}X_i\Big/ \bigvee_{i=1}^{n}X_i$ where $X_1,X_2,\dots,X_n$ are CW-complexes with respective basepoints $x_1,x_2,\dots,x_n$. This construction, in a sense, generalizes the smash product so I wonder if there is a common way to denote this space.
On the other hand, one might want to consider the cell structure of $\displaystyle\prod_{i=1}^{n}X_i\Big/\bigcup_{S\in [n]^2}\prod_{i\in S}X_i\times \prod_{i\notin S}\{x_i\}$ where $[n]^2$ denotes the set of subsets of $\{1,2,\dots ,n\}$ of ordered $2$. You could consider the analogous construction for subsets of order $3$, and so on up to subsets of order $n-1$.
The notation here seems to be getting pretty complicated pretty fast. Is there standard notation and/or terminology for these constructions? If so, I'd be interested in a reference. Even if it's not standard, a "traditional" reference that at least defines some convenient notation for these constructions, would be helpful.