As suggested by Mirko Swirko, you may want to look at indecomposable continua. A continuum is a compact connected Hausdorff space. A continuum is indecomposable if it is not the union of two proper subcontinua.
The following theorem says that we can "almost" disconnect an indecomposable continuum. Specifically, we can write $X$ as the union of two closed sets whose intersection is contained in an given open set.
**Theorem.** If $X$ is an indecomposable continuum then for each open set $U$ there exists two nonempty closed sets $A$ and $B$ with $X=A\cup B$ and $A\cap B\subseteq U$.
**Another theorem.** If $X$ is a _metric_ indecomposable continuum then we can partition $X$ into $2^\omega $ many dense subsets such that any proper closed subset of $X$ intersecting two of these partition sets is not connected!
The simplest example of a metric indecomposable continuum is the Knaster "buckethandle" continuum (google it).