# Thread: closed subsets of a connected space X

1. ## 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

2. ## Re: closed subsets of a connected space X

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.

3. ## Re: closed subsets of a connected space X

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?