Proposition 8.6.2. Let $E$ be a TVS over $K \in \RC$, $M \subset E$ be a subspace, then $E/M$ is Hausdorff if and only if $M$ is closed.
Proof. The space $M$ is closed if and only if
\[M = \bigcap_{V \in \cn(0)}M + V\]
which is equivalent to $E/M$ being Hausdorff.$\square$