weak direct product

《Algebra》 · 第 60 页 · Section I.8「Direct Products and Direct Sums」，Definition 8.3

The (external) weak direct product of a family of groups $\{G_i\mid i\in I\}$, denoted $\prod_{i\in I}^{w}G_i$, is the set of all $f\in\prod_{i\in I}G_i$ such that $f(i)=e_i$, the identity in $G_i$, for all but a finite number of $i\in I$.