This is question from Topology without tears by S. Morris. It reads;
Let $(X, \tau)$ be a topological space with the property that every subset is closed. Prove that it is a discrete space.
I actually think it's not true though, my counter example:
$$let\quad X = \{x, y\}\quad and \quad \tau = \{\emptyset, X\} $$
Cleary $\tau$ is a topology and all it's subsets are closed, but it's not a discrete space. What am I missing?
Best Answer
If every subset of $X$ is closed, then every subset of $X$ is open! Hence $ \tau$ is the discrete topology on $X$. In your example not every subset of $X$ is closed (e.g. $\{x\}$).