Exercise 6, § 4

For each real number $a \geqslant 1$, put

Let $C_a$ denote the set of all points $f_a(t)$ as $t$ runs through $\mathbf{R}$, and let $X \subset \mathbf{R}^2$ be the union of the sets $C_a$ for $a \geqslant 1$ and the lines $D'$ and $D''$, where $D'$ (resp. $D''$) is the set of points $(t, -1)$ (resp. $(t, 1)$) for $t \in \mathbf{R}$.

a) The group $\mathbf{R}$ operates on $X$ according to the law $(s, z) \to s.z$ such that:

(i) $s.(t, -1) = (s+t, -1)$; (ii) $s.(t, 1) = (t-s, 1)$; (iii) $s.f_a(t) = f_a(s+t)$ for $a \geqslant 1$.

Show that $\mathbf{R}$ operates continuously on $X$ and that the four conditions of Proposition 4 are satisfied, but that $X/\mathbf{R}$ is not Hausdorff.

b) Let $S$ be the equivalence relation on $X$ whose classes consist of a single point $z$ if $z \notin D' \cup D''$, and are of the form $(z, -z)$ if $z \in D'$ or $z \in D''$. Let $X'$ be the quotient space $X/S$. The relation $S$ is compatible with the group $\mathbf{R}$ operating on $X$ ($\S$ 2, no. 4); show that $\mathbf{R}$ operates continuously on $X'$, that the four conditions of Proposition 4 are satisfied and that $X'/\mathbf{R}$ is Hausdorff, but that $\mathbf{R}$ does not operate properly on $X'$.*