Proposition 10

If X is a Noetherian space, the set of irreducible components of X (and a fortiori the set of connected components of X) is finite.

It is sufficient to prove that X is a finite union of irreducible closed subsets (no. 1, Proposition 6). Let us show that the principle of Noetherian induction can be applied taking E to be the set of closed subsets of X, ordered by inclusion, and F to be the set of finite unions of irreducible closed subsets. Let Y be a closed subset of X such that every closed subset $\neq Y$ of Y belongs to F. If Y is irreducible, then $Y \in F$ by definition; otherwise, Y is the union of two closed subsets $Y_1,\ Y_2$ which are distinct from Y. Then $Y_1 \in F$ and $Y_2 \in F$ by hypothesis, whence $Y \in F$ by definition of F.

In particular it follows that a Hausdorff Noetherian space is necessarily finite.