Question: Let $Y \subset X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X-Y$, then $Y \cup A$ and $Y \cup B$ are connected.
Attempt at an answer: since $A,B$ form a separation of $X-Y$ then $X-Y = A \cup B$ and $A \cap B = \emptyset$. Then $X = A \cup B \cup Y = (A\cup Y) \cup (B \cup Y)$, but $X$ is connected hence it must be that $(A \cup Y) \cap ( B \cup Y) \neq \emptyset$, but this must imply that $A \cup Y$ and $B \cup Y$ are connected otherwise $X$ could possibly be disconnected if they are not connected. Is this enough, I feel like I a missing some details ( $Y$ is not necessarily an open set in $X$).