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