Exercise 4, § 5

Let $\Gamma$ be the group of homeomorphisms of the quotient space $T$ of $\mathbf{R}$ consisting of the identity and the homeomorphism induced on $T$ by the homeomorphism $x \to 1/2 + x$ of $\mathbf{R}$ onto itself. Let $S$ be the equivalence relation "there exists $\sigma \in \Gamma$ such that $y = \sigma(x)$" on $T$; $S$ is both open and closed (Exercise 3). Show that $T$ has no continuous section with respect to $S$, although both $T$ and $T/S$ are compact, connected and locally connected. Let $A$ be the canonical image in $T$ of the interval $[0,1/2]$ of $\mathbf{R}$. Show that $A$ is locally closed in $T$ but that the quotient space $A/S_A$ is not homeomorphic to the canonical image of $A$ in $T/S$. \*