\begin{definition}[Definition:Irreducible Subset (Topology)]
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A$ be a subset of $S$.
Then $A$ is '''irreducible (subset)''' {{iff}}
:$A$ is non-empty and closed and for all closed subsets $B, C$ of $S$: $A = B \cup C \implies A = B$ or $A = C$
\end{definition}