1. ## Problem about conected spaces

Hello, this is a problem from Munkres: Let X and Y be conected spaces such that $Y\subset X$. If A and B are a separation of X-Y prove that $Y\cup A\,and\,Y\cup B$ are conected.
I can only prove that $Y\cup A\,or\,Y\cup B$ is conected and since I've read from a Spanish transation I wonder whether the problem is correctly formulated.

2. What if $U~\&~V$ is a separation of $A\cup Y?$
We know that because $Y$ is connected we have $Y\subseteq U \text{ or } Y\subseteq V$.
Say $Y\subseteq U$. What does that say about $X=U\cup V\cup B?$

Use similar argument for $B\cup Y$.

3. I'm sorry but I can not see why $U\cup V\cup B$ should be a separation for $X$

4. Originally Posted by facenian
I'm sorry but I can not see why $U\cup V\cup B$ should be a separation for $X$
Let $C=U\cup B$. Then Because $Y\subseteq U$ we have $Y\subseteq C$.
Also it must be the case that $V\subseteq A$.
But $A~\&~B$ are separated sets.
What about $V~\&~C?$
Is this true $V\cup C=X?$

5. You're right, now I see it. Thank you