Exercise 1, Appendix 0

Let $X$ be a Hausdorff topological space on which there is defined an associative law of composition written multiplicatively. Generalize Definition 1 to the case of a family $(x_i)_{i \in I}$ of elements of $X$, whose index set $I$ is linearly ordered. For each finite subset $J$ of $I$, let $p_J$ denote the product $\prod_{i \in J} x_i$ of the sequence $(x_i)_{i \in J}$.

If $X$ is a complete topological group, then a family $(x_i)_{i \in I}$ of points of $X$ is multipliable if and only if, given any neighbourhood $V$ of the identity element $e$ of $X$, there exists a finite subset $J_0$ of $I$ such that, for every finite subset $J$ of $I$ containing $J_0$, we have $p_J(p_{J_0})^{-1} \in V$ [or $(p_{J_0})^{-1} p_J \in V$].