Let $L$ be a subset of $\mathbb{R}^{2}$ and let $N = \mathbb{R}^{2}/L$ be the quotient space obtained by identifying all points in $L$ to a single point. I need to prove that $N$ is Hausdorff $\longleftrightarrow$ $N$ is $T_{1}$ $\longleftrightarrow$ $L$ is a closed subset of $\mathbb{R}^{2}$. I know that a if $N$ is Hausdorff then every finite point set is closed and therefore $N$ satisfies the $T_{1}$ condition. I also know that if $N$ is Hausdorff then for each pair $n_{1},n_{2}$ of distinct points of $N$, there are disjoint neighborhoods $U_{1},U_{2}$ of $n_{1},n_{2}$ respectively.
Thanks