Exercise 25, § 3

Let $G$ be a group endowed with a topology $\mathscr{T}$ with respect to which $G$ is locally compact and the translations $t \to st$ and $t \to ts$ are continuous on $G$ for all $s \in G$. Show that $\mathscr{T}$ is compatible with the group structure of $G$ (Theorem of R. Ellis). [Identify $G$ with the group of left translations of $G$, endowed with the topology of pointwise convergence, so that $G$ is identified with a group of homeomorphisms of the one-point compactification $X$ of $G$, endowed with the topology of pointwise convergence. Using Exercise 24, deduce that on $G$ the topology of pointwise convergence in $X$ is the same as the topology of uniform convergence in $X$ (no. 4, Theorem 3, Corollary 1), and use Proposition 11 of no. 5.]