Cantor intersection theorem for connectedness: a nested intersection of compact connected sets is connected

Let $X$ be a Hausdorff topological space and let

be a decreasing sequence of compact, connected (more precisely: preconnected) subsets of $X$. Then the intersection

is again preconnected.

This is the connectedness companion of Cantor's intersection theorem (which asserts that $K$ is nonempty when every $K_n$ is). Compactness is essential: in the plane the decreasing sequence of closed connected sets $\{(x,y) : x \ge n\}$ has empty — hence connected — intersection, but replacing "compact" by "closed" in general fails, the standard counterexample being a decreasing sequence of closed connected subsets of the plane whose intersection is a pair of disjoint horizontal rays.

The proof is the standard separation argument: if $K = A \sqcup B$ with $A, B$ disjoint, closed and nonempty, then $A$ and $B$ are disjoint compact sets, so by the Hausdorff property they can be surrounded by disjoint open sets $U \supseteq A$ and $V \supseteq B$. The compact sets $K_n \setminus (U \cup V)$ decrease and have empty intersection, so one of them is empty; that $K_n$ is then split by $U$ and $V$ into two nonempty relatively open pieces, contradicting its connectedness.

Preconnectedness (rather than connectedness) is the right formulation because the intersection may be empty.