Irreducible components of a topological space

Let $(X,\mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), and let irreducible subsets be as in Irreducible topological spaces and irreducible subsets in the subspace topology.

An irreducible component of $X$ is an irreducible subset $C\subseteq X$ which is maximal among the irreducible subsets of $X$ under inclusion (Maximal element and greatest element): if $D\subseteq X$ is irreducible and $C\subseteq D$, then $D=C$.

The empty space has no irreducible components, because an irreducible space is required to be nonempty; a one-point space has exactly one irreducible component, namely the point itself, because the only nonempty subset of a one-point space is the space itself. Assuming the Axiom of Choice (The Axiom of Choice), for a nonempty space the irreducible components exist, they are closed subsets whose union is $X$, every irreducible subset of $X$ is contained in an irreducible component, and $X$ is irreducible exactly when $X$ is its own unique irreducible component. A Noetherian space has only finitely many irreducible components. These statements are proved under that hypothesis in Existence and basic properties of irreducible components and A Noetherian space is a finite union of irreducible closed subsets.