Hartshorne Exercise II.3.17 Noetherian Induction

Question

I'm confused by the Noetherian Induction exercise in Hartshorne.

Let $X$ be a Noetherian topological space, and let $\mathscr{P}$ be a property of closed subsets of $X$. Assume that for any closed subset $Y$ of $X$, if $\mathscr{P}$ holds for every proper closed subset of $Y$, then $\mathscr{P}$ holds for $Y$. (In particular, $\mathscr{P}$ must hold for the empty set.) Then $\mathscr{P}$ holds for $X$.

Using Zorn's Lemma, we can say that the set $\Sigma$ of closed subsets of $Y$ where $\mathscr{P}$ holds has a maximal element, but how do we know that the maximal element is $X$? I want to say that we can take the union of all closed subsets in $\Sigma$ and that should be $X$ but why is the union of all closed subsets in $\Sigma$ equal to $X$?

Thanks!

Answer 1

Consider the set $S$ of closed subsets that do not satisfy $\mathscr{P}$.

Assume that $S$ is not empty. Then you can find an infinite descending chain in $S$ as follows:

for any closed subset $Y$ of $X$, if $\mathscr{P}$ holds for every proper closed subset of $Y$, then $\mathscr{P}$ holds for $Y$

can be rewritten by contrapositive

for any element $Y$ of $S$, there is a proper closed subset of $Y$ that also lies in $S$

But this contradicts $X$ being Noetherian.