I'm struggling with the proof that says a Noetherian topological space $X$ is the finite union of closed irreducible subsets. In particular with this part:
First observe that every nonempty set of closed subsets of $X$ has a minimal element, since otherwise it would contain an infinite strictly descending chain.
I get that a chain $Y_1\supsetneq Y_2\supsetneq\ldots$ should terminate by the Noetherness of $X$. But why is it not possible to have a nonempty set of closed subsets, in which $Y_i\nsupseteq Y_j$ for all $i,j$?