Irreducible space

Definition

A nonempty topological space $X$ is irreducible if it cannot be written as the union of two proper closed subsets. Equivalently, every two nonempty open subsets of $X$ intersect.

A subset $Z\subseteq X$ is irreducible if it is irreducible with its subspace topology. An irreducible component of $X$ is a maximal irreducible subset; every irreducible component is closed.

If a point $\eta\in X$ has closure equal to $X$, then $X$ is irreducible and $\eta$ is its generic point. In particular, if $A$ is an integral domain, then the prime spectrum $\operatorname{Spec}A$ is irreducible, with generic point $(0)$.