= Connected union with a connected hub
Suppose <connected subsets> $U_j$ all meet one <connected> member $U_1$. A separation of their union would place $U_1$ entirely on one side. Each $U_j$ is also entirely on one side and intersects $U_1$, so every $U_j$ lies on the same side, contradicting nonempty separation. Thus the union is <connected>. If the subsets are <path-connected>, join two points by a <path> to $U_1$, a <path> inside $U_1$, and a <path> back out. This needs pairwise contact with the hub, not one point shared by every subset.
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