I was recently attempting proving a conjecture of mine about the existence of certain minimal nonempty closed sets in a topological space. I opted to use Zorn's Lemma, and the proof would go through except for the following problem:
Question: In a topological space $X$, let $\mathcal{A} = \{A_\lambda : \lambda\in\Lambda\} \subseteq \mathcal{P}(X)$ be a chain of nonempty closed sets, under $\subseteq$. When is $A = \bigcap_{\lambda\in\Lambda} A_\lambda$ nonempty?
One possible avenue: This question appears similar to the Cantor Intersection Theorem and its variants. However, that theorem requires the chain to be countable, and it also requires some additional conditions on the members of the chain. Can we get a general version when $\Lambda$ is uncountable? Perhaps there is a way involving a well-ordering on $\Lambda$?
The proof of Cantor's theorem presented in Wikipedia is suboptimal (anybody can edit a Wikipedia article, no matter how little do they know about the subject). Here is a more general proof that answers (I hope) your question which is a bit open-ended.
Theorem. Suppose that $K$ is a compact topological space and $\{A_j: j\in J\}$ is a chain of nonempty closed subsets of $K$ (ordered with respect to the inclusion. Then the intersection $$ \bigcap_{j\in J} A_j $$ is nonempty.
Proof. It is useful to rewrite the definition of compactness using closed subsets:
A topological space $X$ is compact if and only if for every family $\{C_i: i\in I\}$ of closed subsets of $X$ with empty intersection, there exists a finite (nonempty) subset $F\subset I$ such that $$ \bigcap_{i\in F} C_i=\emptyset. $$
Now, back to the proof of the theorem. Suppose, for the sake of a contradiction, that $$ \bigcap_{j\in J} A_j=\emptyset. $$ Then, by compactness of $K$, there exists a finite subset $F\subset J$ such that $$ \bigcap_{j\in F} A_j=\emptyset. $$ Let $A_i$ be the smallest among $A_j, j\in F$. Then $$ A_i=\bigcap_{j\in F} A_j=\emptyset. $$ A contradiction (since we assumed that each $A_j$ is nonempty). qed