Connectedness of an intersection from connected finite intersections (${\bigcap }_{j\in J}{K}_j\text{ nonempty and connected for finite }J⟹{\bigcap }_i{K}_i\text{ compact and connected}$)

Let $X$ be a compact Hausdorff space and let $K_i$ be compact subsets such that all finite intersections are nonempty and connected. Their total intersection $K$ is nonempty by the finite intersection characterization of compactness, and compact because it is closed in $X$.

If $K$ were not connected, its separation of a topological space would divide it into two nonempty compact subsets. Separation of disjoint compact subsets gives disjoint ambient open sets $U,V$ containing the two pieces. The family consisting of $X\setminus (U\cup V)$ and all $K_i$ has empty total intersection, so some finite subfamily has empty intersection. Hence a finite intersection $K_J$ of the $K_i$ lies in $U\cup V$. Since $K\subseteq K_J$ meets both open sets, they disconnect $K_J$, a contradiction. The hypothesis on all finite intersections is stronger than requiring each individual $K_i$ to be connected.

Use the arcs and points of the noncompact counterexample to the Šura-Bura lemma, and add the central ray $\{(1/2,t):t\ge 1\}$ to form $X$. Define $C_n=\{p,q\}\cup {\bigcup }_{m\ge n}{A}_m\cup \{(1/2,t):t\ge n\}$. Each graph tail is connected because its arcs meet the ray. Its relative closure in $X$ adds just $p,q$, so each $C_n$ is closed and connected. These sets decrease and have intersection $\{p,q\}$, which is compact and disconnected. The ambient $X=C_1$ is connected and Hausdorff but noncompact.