Definition:Irreducible Subset (Topology)

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A$ be a subset of $S$.

Then $A$ is irreducible (subset)
 * $A$ is non-empty and closed and for all closed subsets $B, C$ of $S$: $A = B \cup C \implies A = B$ or $A = C$