Exercise 17, § 3

a) Show that if $X$ is a locally compact, locally connected space, then the topology induced on the group $\Gamma'$ of all homeomorphisms of $X$ by the compact-open topology coincides with the topology $\tilde{\mathscr{O}}_\beta$ defined in no. 5 [use Exercise 15 b) and Proposition 12 of no. 5].

b) Let $X$ be the locally compact subspace of $\mathbf{R}$ consisting of the points $0$ and $2^n (n \in \mathbf{Z})$. If $\Gamma$ is the group of all homeomorphisms of $X$, show that the topology induced on $\Gamma$ by the compact-open topology is not compatible with the group structure of $\Gamma$.