 Decreasing closed connected sets with disconnected compact intersection

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.