Understanding connected point sets.

Prove whether the following is true or false. If it is false give a counter example.

If F is a collection, each element of which is a connected point set and each point set in F contains a limit point of every other point set in F, then UF is connected.

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... generality, assume it lies entirely within C. Now D must be non-empty, so there must be an F_beta in F such that F_beta intersect D is non-empty. Since F_beta is connected, it must lie entirely within D. If a set is connected, it's ...