Closure of a point is irreducible ($\overline{\{x\}}$)

In any topological space, the closure $C$ of a point $x$ is an irreducible closed subset. If $C=A\cup B$ with $A,B$ closed in $C$, then both are closed subsets of the ambient topological space. One contains $x$, so it contains all of $C$ by the definition of closure. Equivalently, $x$ is a generic point of $C$. This does not assert that every irreducible closed subset of an arbitrary topological space has a generic point.