If quotient by a closed normal subgroup is discrete

Question

Suppose that $T$ is a topological group and $K$ is a closed normal subgroup. Is $T/K$ a discrete space?

I think it is, since $tK$ for all $t\in T$ is closed, i.e. every singleton $\{tK\}$ is closed. But why most books state that such a quotient is Hausdorff if it is true that it is discrete?

Answer 1

$\mathbb{R}/{\mathbb{Z}}$ is just the circle group $S^1$, far from discrete.

The fact that all cosets are closed (which is true) just says that the quotient is $T_1$ and thus Tychonoff (as a topological group as well). In my example $K$ is discrete and the resulting quotient is compact Hausdorff and metrisable.