Corollary

Let X be a compact connected topological space, let N be the set of points of X whose complement is connected. The set X is the unique connected and compact subset of X which contains N.

Let S be a connected and compact subset of X such that $N\subset S$. Suppose that S $= X\not$ and let $x\in X-S$. By hypothesis, X $-\{x\}$ is not connected; there therefore exist open and closed subsets U and V of X $-\{x\}$, disjoint and nonempty, such that X $-\{x\}= U\cup V$. We may suppose that $S\subset V$. We have $\overline{U}= U\cup \{x\}$ and $V = V\cup \{x\}$ and these spaces are connected (III, p. 278, prop. 17, a)). By assertion c) of this proposition, there exists $y\in U$ such that X $-\{y\}$ is connected, which contradicts the inclusions $N\subset S\subset V$.