Irreducible closed subset

A nonempty closed subset which, with its induced topology, is an irreducible topological space. In an affine algebraic set these subsets correspond, with reversed inclusion, to prime ideals of the coordinate ring.

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.