Let $X = [-1,1] \times \{0, 1 \} \subset \mathbb{R}^2$ with the induced topology on $\mathbb{R}^2$. Then $X$ is a Hausdorff space as a subspace of a Hausdorff space.
The question is now, if
$Y = X \setminus_{\sim}$ with $\sim$ induced by $(t,0) \sim (t,1)$ $\forall t \in [-1,- 1/2] \cup [1/2, 1]$ is also Hausdorff ?
I think yes, but I do not see how to prove this statement.
Let $p,q\in X/\sim$ with $p\neq q$, and $\pi: X \to X/\sim$ the canonical map. Observe that $(X/\sim)\setminus \{ [(-1/2,0)],[1/2,0] \}$ can be cover with the disjoint open sets of $X/\sim$:
Each $X_i$ is Hausdorff because $\pi^{-1}(X_i)$ is open and Hausdorff, and $\pi^{-1}(p)$ is finite for each $p\in X_i$. So we only need to prove the Hausdorff property for $[(-1/2,0)]$ with respect to al $X/\sim$ and the same for $[1/2,0]$. This easy again because the preimage by $\pi$ of these points are finite.