closed subsets of a connected space X

**rqeeb** · October 11th 2011, 07:09 PM

Let $A\& B$ be closed subsets of a connected space $X$ such that $X = A \cup B$ .
How to show $A\& B$ are connected, if $A \cap B$ is connected.

I traied this:
Suppose $B$ is not connected and let $B = S \cup T$ be sepration of $B$ , then,
$A \cap B$ must lie in $S$ (say) So,
$X = \left( {A \cup S} \right) \cup T$
form separation of x
$\Rightarrow \Leftarrow$

is this right

**Plato** · October 12th 2011, 02:48 AM

Originally Posted by rqeeb

Let $A\& B$ be closed subsets of a connected space $X$ such that $X = A \cup B$ .
How to show $A\& B$ are connected, if $A \cap B$ is connected. I traied this:
Suppose $B$ is not connected and let $B = S \cup T$ be sepration of $B$ , then,
$A \cap B$ must lie in $S$ (say) So,
$X = \left( {A \cup S} \right) \cup T$
form separation of x
$\Rightarrow \Leftarrow$

Now say that likewise $A$ must be connected.

**rqeeb** · October 12th 2011, 05:10 AM

Originally Posted by Plato

Now say that likewise $A$ must be connected.

but now the problem is: To show AUS and S are spen in X
we know that S & T are open in B, and B is closed in X. Bur how to show they r open in X?

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