Irreducible space - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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

Equivalent characterization

Equivalently, every two nonempty open subsets of $X$ intersect.

Subsets and components

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.

Generic points

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 $(0)$ .