Show a metrix space is connected iff for every non-empty proper subset the boundary is non-empty Show a metric space X is connected iff \forall non-empty proper subset A \subset X the boundary (set of points in X whose neighborhoods contain points from both A and the complement) is non-empty.

If $X$ is disconnected, so $X = A \cup B$ where $A$ and $B$ are disjoint non-empty closed (and open) subsets. What is $\operatorname{Bd}(A)$?

Suppose $A$ is non-empty and suppose $\operatorname{Bd}(A) = \emptyset$; as the boundary of $A$ is the difference between closure and interior of $A$, we have that these coincide and so $A$ is clopen. So $X = A \cup \ldots $ proves $X$ is disconnected.