Problem about conected spaces

**facenian** · August 13th 2010, 06:37 AM

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.

**Plato** · August 13th 2010, 07:16 AM

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 $B\cup Y$ .

**facenian** · August 13th 2010, 09:31 AM

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

**Plato** · August 13th 2010, 10:02 AM

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 $V\cup C=X?$

**facenian** · August 13th 2010, 10:58 AM

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

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