Discrete topology

The discrete topology on a set $X$ is the topology such that the open sets are the subsets of $X$, hence $\tau:=\mathcal{P}(X)$.

Remarks

• In particular, all singletons $\{x\}$ are both open and closed since $\{x\}\subset X$ and $\{x\}$ is the complement of $X\backslash\{x\}\subset X$.