Here's what I know:
When is a (metric) space disconnected?
Consider a metric space $(M,d)$. $M$ is disconnected if there exist non-empty disjoint open subsets $A,B \subset M$ such that $A\cup B = M$. $\{A,B\}$ is called a disconnection of $M$.
When is a subset of $M$ disconnected?$\color{red}{^1}$
Consider $E\subset M$. $E$ is disconnected if there exists non-empty disjoint open subsets $A,B \subset M$ such that $E\subset A\cup B$, $A\cap E\neq \varnothing$ and $B\cap E\neq \varnothing$.
I have seen that using the first definition, people often say that $M$ is disconnected if there exist non-empty disjoint closed sets such that $A\cup B = M$. This makes sense here, since $A^c = B$ and $B^c = A$, so if $A,B$ are open, they are also closed.
What about the second definition though, i.e. when is $E$ disconnected? Can we come up with something in terms of closed sets here?
My problem is that in the second definition, $A^c$ is not necessarily $B$ – so nothing useful comes out of that apparently. Could someone help me out here, and better clarify notions of connectedness/disconnectedness for me? Thank you very much.
Footnote:
$\color{red}{1.}$ In Carothers' Real Analysis, this characterization of disconnectedness of subsets of $M$ has been motivated by the relative metric, and notions of open sets in $E$.
Best Answer
A subset $E \subseteq M$ is disconnected exactly when $(E,d)$ in the relative metric (or as a topologist would say, the subspace topology) is disconnected.
We can also use the same reformulation as you have given but with closed sets instead. So
The reason that this works is that $A \cap E$ and $B \cap E$ are then open and closed in $(E,d)$ just as in the open $A$,$B$ case.
Sidenote: for general (non-metric) spaces we don't demand that $A \cap B = \emptyset$ but $A \cap B \cap E = \emptyset$, to avoid weird examples). So the sets need only be disjoint on $E$, not in the whole space.