Rudin functional analysis, theorem 3.23. Finite intersection property

Question

This is a follow up question to this one.

Continuing through the proof there's a subtlety I don't get, but I think it would be educational to understand to improve my background in general topology.

Choose some $S \in \mathcal{P}$. Let $\mathcal{P}'$ be the collection of all members of $\mathcal{P}$ that are subsets of $S$. Since $S \in \mathcal{P}'$, $\mathcal{P}'$ is not empty. Partially order $\mathcal{P}'$ by set inclusion, let $\Omega$ be a maximal totally ordered subcollection of $\mathcal{P}'$, and let $M$ be the intersection of all members of $\Omega$. Since $\Omega$ is a collection of compact sets with the finite intersection property, $M \neq \emptyset$.

Now I'm referring to theorem in this page,

Is such theorem the one used implicitly by Rudin to assess the finite intersection property?

Also… the proof continues as

The maximality of $\Omega$ implies that no proper subset of $M$ belongs to $\mathcal{P}$.

Not sure I get this bit as well… how exactly is maximality used to asses that no proper subset of $M$ doesn't belong to $\mathcal{P}$

Answer 1

• Rudin uses Zorn's lemma to prove the existence of $\Omega$.
• The finite intersection property is a known fact of compact sets X: Any collection of closed subsets of X with the finite intersection property has nonempty intersection.