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

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

Use similar argument for $\displaystyle B\cup Y$.

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

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

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