$Y \subset X$ where $X$ and $Y$ are connected but $X-Y$ is separated by $A$ and $B$

Question

It is known that $Y \cup A$ and $Y \cup B$ are connected given that $X$ is a connected space, $Y$ is a connected subspace of $X$, and that $X-Y$ is separated by $A$ and $B$. Is it true that $A$ and $B$ are connected as well?

Answer 1

No, this is false. Let $X \subset \Bbb{R}^2 $ be the union of two closed discs $D_1, D_2$ whose boundaries intersect in a single point $p$. Let $L$ be the line connecting $p$ to the centers of both discs, and $Y = X \cap L$. If you let $A = D_1 /Y$, $B = D_2/Y$ then $Y$, $Y \cup A$ and $Y \cup B$ are all connected and $X/Y$ is separated by $A$ and $B$, but both $A$ and $B$ are not connected.