Exercise 1, Appendix 2

Let $E$ be a separated topological space, in which an associative composition law, denoted multiplicatively, has been defined. Generalize Def. 1 of IX, p. 79, to the case of a family $(x_i)_{i \in I}$ of elements of E, whose set of indices I is totally ordered; for every finite subset J of I, the product $p_J$ $\prod_{i \in J} x_i$ of the sequence $(x_i)_{i \in J}$ will be denoted by $p_J$.

If E is a complete topological group, in order that a family $(x_i)_{i \in I}$ of points of E be multipliable, it is necessary and sufficient that for every neighbourhood V of the identity element e of E, there exist a finite subset $J_0$ of I such that, for every finite subset J of I containing $J_0$, one has $p_J(p_{J_0})^{-1} \in V$ (or $(p_{J_0})^{-1} p_J \in V$).