Proposition 25

Let $(G_i)_{i \in I}$ be a family of topological groups, and let $H$ be the normal subgroup of $G = \prod_{i \in I} G_i$ consisting of all $x = (x_i)$ such that the $x_i$ are equal to the identity element $e_i$ of $G_i$ except for a finite number of indices. Then the subgroup $H$ is dense in $G$.

This is a particular case of Chapter I, § 4, no. 3, Proposition 8.

Let $(X_i)_{i \in I}$ be a family of topological spaces, and for each $i \in I$ let $G_i$ be a topological group operating continuously on $X_i$. It is clear that the product group $G = \prod_{i \in I} G_i$ then operates continuously on the product space $X = \prod_{i \in I} X_i$ according to the law

(Chapter I, § 4, no. 1, Proposition 1, Corollary 1, and Proposition 2). Moreover the orbit under $G$ of a point $x = (x_i)$ of $X$ is the product of the orbits of the $x_i$ (with respect to the groups $G_i$). Let $\varphi_i$ be the canonical mapping of $X_i$ onto $X_i/G_i$, and let $\varphi = (\varphi_i)$ be the product mapping of $X$ onto $\prod_{i \in I} (X_i/G_i)$; then the preceding remark shows that the bijection canonically associated with $\varphi$ maps the orbit space $X/G$ onto $\prod_{i \in I} (X_i/G_i)$. Moreover: