= 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 $\{(1/2,t):t\geq1\}$ to form $X$. Define $C_n=\{p,q\}\cup\bigcup_{m\geq n}A_m\cup\{(1/2,t):t\geq n\}$. 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.
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