= Connectedness of an intersection from connected finite intersections
{title2=$\bigcap_{j\in J}K_j\text{ nonempty and connected for finite }J\ \Longrightarrow\ \bigcap_iK_i\text{ compact and connected}$}
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>.
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