Corollary

Let X be a simply connected topological space and let A be a connected subset of X. If the complement of A is connected, its boundary is also connected.

Let $X_1$ and $X_2$ be the closures of A and of X-A respectively. The sets $X_1$ and $X_2$ are closed and connected (TG, I, p. 81, prop. 1); one has $X_1\cup X_2= X$ and their intersection $X_1\cap X_2$ is equal to the boundary of A. It is then enough to apply proposition 7.