Let $X$ and $Y$ be connected, and let $Y \subseteq X$. If $A$ and $B$ are two non-empty, disjoint open sets (open in the subspace $X-Y$) whose union is $X-Y$, or in other words if $A$ and $B$ form a separation of $X-Y$, then how to prove that $Y \cup A$ and $Y \cup B$ are connected?
HINT: suppose $C$ and $D$ are open in $X$ and form a separation of $Y \cup A$. Then, since $Y$ is connected, it must lie entirely in one of $C$ or $D$, suppose it is $C$. But then $D \cap A$ and $C \cup B$ form a separation of $X$.
What's left to prove is that this works even when $B$ and $A$ are not open in $X$.