= Solution
Put $K=\bigcap_iF_i$. If some $F_i$ is empty, so is $K$, which is <connected>. Otherwise the <compact> sets are closed in the Hausdorff ambient space, and can all be considered as nested nonempty closed subsets of the <compact> space $F_0$. Part (a)(ii) gives $K\ne\varnothing$, and $K$ is <compact>.
Suppose $K$ were disconnected. Part (c)(iii) gives disjoint ambient <open sets> $U,V$ covering $K$, both meeting it. Part (a)(iii), applied within $F_0$ to $U\cup V$, gives $F_N\subseteq U\cup V$ for some $N$. Both $U$ and $V$ meet $F_N$ because they meet $K\subseteq F_N$. Their intersections with $F_N$ would therefore disconnect $F_N$, a contradiction. Thus
$$
\boxed{K\text{ is connected}.}
$$
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