sober space

Let $X$ be a topological space. A subset $A$ of $X$ is said to be irreducible if whenever $A\subseteq B\cup C$ with $B,C$ closed, we have $A\subseteq B$ or $A\subseteq C$. Any singleton and its closure are irreducible. More generally, the closure of an irreducible set is irreducible.

A topological space $X$ is called a sober space if every irreducible closed subset is the closure of some unique point in $X$.

Remarks.

• For any sober space, the closure of a point determines the point. In other words, $\mathrm{cl}(x)=\mathrm{cl}(y)$ implies $x=y$.

• A space is sober iff the closure of every irreducible set is the closure of a unique point.

• Any sober space is T0.

• Any Hausdorff space is sober.

• A closed subspace of a sober space is sober.

• Any product of sober spaces is sober.