Results on intersections of compact sets in non-metrizable spaces We know that if $X$ is a Hausdorff topological space and $C_0 \supseteq C_1 \supseteq ...$ are nested, non-empty compact sets, then $\bigcap_{i \in \mathbb{N}} C_i \neq \emptyset$, this is Cantor's Intersection Theorem. However, this gives us no information about the "size" of $\bigcap_{i \in \mathbb{N}} C_i$. For instance, we could take $X= \mathbb{R}$ and take $C_i= [1- \frac{1}{i+1}, 1+ \frac{1}{1+i}]$ and get $\bigcap_{i \in \mathbb{N}} C_i= \left\{1 \right\}$, but we could also give an expression for $C_i$ such that $\bigcap_{i \in \mathbb{N}} C_i$ yields the Cantor Set and hence is uncountable. I am interested in statements of the following form:

"For a non-metrizable topological space $X$ and compact, non-empty $C_i \subseteq X$, if $C_i$ each have Property (1), then $\bigcap_{i \in \mathbb{N}} C_i$ contains exactly one element."

I emphasize "non-metrizable" here because there are analogous statements for metrizable spaces and the specific spaces I'm looking at are never metrizable. Does anyone know of any theorems of the above form?

Also, obviously if $|C_i|=1$ for each $i$ then the statement holds, so anything other than that would be better.
We have the follow related theorems:

Compact Subspace of a Hausdorff space is Closed: https://proofwiki.org/wiki/Compact_Subspace_of_Hausdorff_Space_is_Closed

Closed Subset of Compact Space is Compact https://proofwiki.org/wiki/Closed_Subspace_of_Compact_Space_is_Compact

Not a complete answer, but you can probably used these to help you with this question.
We have the follow related theorems:

Compact Subspace of a Hausdorff space is Closed: https://proofwiki.org/wiki/Compact_Subspace_of_Hausdorff_Space_is_Closed

Closed Subset of Compact Space is Compact https://proofwiki.org/wiki/Closed_Subspace_of_Compact_Space_is_Compact

Not a complete answer, but you can probably used these to help you with this question.