nLab
irreducible closed subspace

Given a topological space $X$, a closed subspace $F$ of $X$ is irreducible if there are exactly two ways to express $F$ as a union of two closed subspaces: $F=\varnothing \cup F$ and $F=F\cup \varnothing$. In other words, $F$ must be inhabited (so that these two ways are distinct) but it must be impossible to express $F$ as a union of two inhabited closed subspaces.

Note that the closure of any point of $X$ is an irreducible closed subspace. $X$ is sober if and only if every irreducible closed subspace is the closure of a unique point of $X$. In general, the irreducible closed subspaces of $X$ correspond to the points of the topological locale $\Omega (X)$.

The theory of irreducible closed subspaces is not useful in constructive mathematics; instead, one should use the completely prime filters on the frame of open subspaces of $X$ (which are the points of $\Omega (X)$ by definition).