Exercise 20, § 3

Let $X$ be the locally compact subspace of $\mathbf{R}^2$ consisting of the lines $y = 0, y = 1/n \ (n \geq 1)$, and let $G$ be the group of all homeomorphisms $u$ of $X$ whose restriction to each of the lines $y = 0, y = 1/n$ is a translation of the form $(x, y) \to (x + a_y, y)$. Show that $G$ satisfies the conditions of Theorem 4 of no. 5, but that the topologies of compact and pointwise convergence on $G$ are distinct.