closed subsets of a connected space X

Let $\displaystyle A\& B$ be closed subsets of a connected space $\displaystyle X$ such that $\displaystyle X = A \cup B$.

How to show $\displaystyle A\& B$ are connected, if $\displaystyle A \cap B$ is connected.

I traied this:

Suppose $\displaystyle B$ is not connected and let $\displaystyle B = S \cup T$ be sepration of $\displaystyle B$, then,

$\displaystyle A \cap B$ must lie in $\displaystyle S$ (say) So,

$\displaystyle X = \left( {A \cup S} \right) \cup T$

form separation of x

$\displaystyle \Rightarrow \Leftarrow $

is this right

Re: closed subsets of a connected space X

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Originally Posted by

**rqeeb** Let $\displaystyle A\& B$ be closed subsets of a connected space $\displaystyle X$ such that $\displaystyle X = A \cup B$.

How to show $\displaystyle A\& B$ are connected, if $\displaystyle A \cap B$ is connected. I traied this:

Suppose $\displaystyle B$ is not connected and let $\displaystyle B = S \cup T$ be sepration of $\displaystyle B$, then,

$\displaystyle A \cap B$ must lie in $\displaystyle S$ (say) So,

$\displaystyle X = \left( {A \cup S} \right) \cup T$

form separation of x

$\displaystyle \Rightarrow \Leftarrow $

Now say that likewise $\displaystyle A$ must be connected.

Re: closed subsets of a connected space X

Quote:

Originally Posted by

**Plato** Now say that likewise $\displaystyle A$ 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?