Exercise 23, § 2

Consider, on the product set of a family $(G_i)_{i \in I}$ of topological groups, the group structure which is the product of the group structures of the $G_i$ and the topology generated by the set of products $\prod_{i \in I} A_i$, where $A_i$ is open in $G_i$ for each $i \in I$ and where the set of $i \in I$ such that $A_i \neq G_i$ has cardinal $< c$, where $c$ is a given infinite cardinal (Chapter I, § 4, Exercise 9). Show that this topology is compatible with the group structure of $G$. Hence construct an example of a non-discrete Hausdorff topological group in which every compact subset is finite (cf. Chapter I, § 9, Exercise 4).