Let $X$ be two disjoint copies of $\mathbb{R}$, that is say $X = (\{a\} \times \mathbb{R}) \cup (\{b\} \times \mathbb{R})$ for real numbers $a<b$ and consider X as a subspace of $\mathbb{R} \times \mathbb{R}$. Define an equivalence relation by $a \times t \sim b \times t$ for all $t \neq 0$. Show that the quotient space $X^{*}$ determined by this equivalence relation is locally Hausdorff, but not Hausdorff.
Definition of Locally Hausdorff: For any $x$ in $X$ there is a neighborhood $U$ of $x$ such that $U$, with the subspace topology, is Hausdorff.