Proposition 6

Let $X$ be a topological space and $(P_i)_{1 \leq i \leq n}$ ajinite covering of $X$ consisting & irreducible closed sets. Then the irreducible components of $X$ are the maximal elements (with respect to inclusion) of the set of $P_i$.

We may restrict ourselves to the case where no two $P_i$ are comparable. If $E$ is an irreducible subset of $X$, then $E \subset \bigcup_{i=1}^n P_i$, and hence $E$ is contained in one of the closed sets $P_i$; this proves that the $P_i$ are the only maximal irreducible subsets of $X$.