The smash product of the family $\smash {\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}}$ is the pointed set $\bigwedge _{i\in I}X_{i}$ consisting of:
- The Underlying Set. The set $\bigwedge _{i\in I}X_{i}$ defined by
\[ \bigwedge _{i\in I}X_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\prod _{i\in I}X_{i}\webright )/\mathord {\sim }, \]
where $\mathord {\sim }$ is the equivalence relation on $\prod _{i\in I}X_{i}$ obtained by declaring
\[ \webleft (x_{i}\webright )_{i\in I} \sim \webleft (y_{i}\webright )_{i\in I} \]if there exist $i_{0}\in I$ such that $x_{i_{0}}=x_{0}$ and $y_{i_{0}}=y_{0}$, for each $\webleft (x_{i}\webright )_{i\in I},\webleft (y_{i}\webright )_{i\in I}\in \prod _{i\in I}X_{i}$.
- The Basepoint. The element $\webleft [\webleft (x_{0}\webright )_{i\in I}\webright ]$ of $\bigwedge _{i\in I}X_{i}$.
