Proving that a space is disconnected 
Show that a subspace $T$ of a topological space $S$ is disconnected iff there are nonempty sets $A,B \subset T$ such that $T= A\cup B$ and $\overline{A} \cap B = A \cap \overline{B} = \emptyset$. Where the closure is taken in $S$.

I've used this relatively simple proof for many of these slightly different types of questions so I was wondering if it's the right method. It seems pretty good, except for the 'where the closure is taken in $S$ part'. 
$T$ is disconnected if and only if there exists a partition $A,B \subset T$ such that $T = A \cup B$ and $A \cap B = \emptyset$. Also, $A$ and $B$ are both open and closed therefore $\overline{A} = A$ and $\overline{B} = B$. The result follows. 

It looks fine to me, in particular because as $\,A\subset \overline{A}\Longrightarrow A\cap B\subset \overline{A}\cap B\,$ , so if the rightmost intersection is empty then also the leftmost one is, which is the usual definition
