Question about Connected point sets

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

If M is a connected point set, the cl(M) is connected.

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Solution Preview C and D are disjoint, non-empty, open subsets of cl(M). Now M is connected and is a subset of cl(M). Therefore by "Lemma A", M must lie entirely within either C or D. Suppose M is a subset of C. Then cl(M) is a subset of cl(C). ...