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**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 $