This question is similar to exercises in munkres 23-#12 : "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"
So I tried to similarly.
Let $A,B$ is separation of $X-C$. Since $Y$ is connected, $Y\subset A$ or $Y \subset B$. Suppose $Y \subset A$.
No limit point of $A$ can be $B$, and no limit point of $C$ and be $B$ so that $A \cup C$ is closed, and $B$ is open.
And I tried to show that no limit point of $B$ can be $A$ and $C$, but I didn't.
That is, I want to show that $B$ is open in $X$. How does it work?