Let $\displaystyle X$ be a connected topological space and $\displaystyle A \subset X$ connected. Suppose that $\displaystyle A^c = M \cup N $ with $\displaystyle M$ and $\displaystyle N$ separated (i.e. $\displaystyle cl(M) \cap N = \emptyset $ and $\displaystyle cl(N) \cap M = \emptyset $). Prove that $\displaystyle A \cup M$ and $\displaystyle A \cup N$ are connected.

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