Let $E \subset S$. Suppose $E$ is not connected. Then in the induced topology of $E$ relative to $S$, $E$ and $\emptyset$ are not the only clopen sets.
I can show using the above definition that if $E$ is not connected, it can be written as the union of sets $A$ and $B$ such that $A$ and $B$ are separated in $E$.
However, is it necessarily true that $E$ can be written as the union of sets that are separated in $S$? If not, can you give a counter example?
My definition of a connected set is: A set $S$ is connected if and only if the only subsets in it that are clopen are the set $S$ and $\emptyset$. A subset $E \subset S$ is connected if and only if only $E$ and $\emptyset$ are clopen in the induced topology of $E$ relative to $S$.
Let $E$ be a union of sets $A$, $B$ separated in $E$. If they are not separated in $S$, there exists e.g. $a\in A$ such that $a\in\overline{\!B\,}^S$. If $U$ is an open $E$-neighborhood of $a$, there's an open $S$-neighborhood $V$ of $a$ such that $U=V\cap E$. Since $V\cap B\ne\varnothing$, also $U\cap B=(V\cap E)\cap B=V\cap(E\cap B)=V\cap B\ne\varnothing$. Therefore $a\in\overline{\!B\,}^E$, which contradicts $A$ and $B$ being separated, so they are separated in $S$ after all.