Exercise 7, § 8

Let $X$ be the interval $[ -1, 1 ]$ in $\mathbf{R}$. Consider the following equivalence relation $S$ on $X$: if $x \neq \pm 1$, the equivalence class of $x$ consists of $x$ and $-x$; the equivalence class of $1$ (resp. $-1$) consists of $1$ (resp. $-1$) alone. Show that $S$ is open and that the quotient space $X/S$ is accessible but not Hausdorff. Show also that every point of $X/S$ has a neighbourhood in $X/S$ which is a regular subspace. \*