Question is :
Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that $Y\cup A$ and $Y\cup B$ are connected.
What i have tried is :
Suppose $Y\cup A$ has separation say $C\cup D$ then all i could see is that
Either $Y\subset C$ or $Y\subset D$ as $Y$ is connected and $
C\cup D$ is separation for a subset that contains $Y$.
Without loss of generality we could assume $Y\subset C$
As $Y\cup A=C\cup D$ and $Y\subset C$ i can say $D\subset A$ (I do not know how does this help)
I have $X-Y=A\cup B$ with $A\cap B=\emptyset$
I have not used connectedness of $X$ till now, So i thought of using that and end up with following :
$X-Y=A\cup B\Rightarrow X=Y\cup A\cup B=(Y\cup A)\cup (Y\cup B)$
I do not know what to conclude from this…
I would be thankful if some one can help me by giving some "hints"
Thank you
Best Answer
As $A,B$ form a separation for $X-Y$ and $X$ is connected, $Y$ must be non-empty.
You suppose that $Y\cup A$ has a separation $C\cup D$. Then $Y$, being connected, is a subset of $C$
Now $A$ is clopen in $X-Y$, that means there are sets $U$ open and $K$ closed, such that $U-Y=K-Y=A$. You know that $D\subseteq A$ is clopen in $Y\cup A$. Can you show that $D$ is open in $U$ and closed in $K$?