Continuum = compact connected set.
Suppose that $U$ and $V$ are nonempty disjoint open subsets of $[0,1]^2$. Is there necessarily a continuum $K\subseteq [0,1]^2$ that divides $U$ and $V$? More precisely, is there a continuum $K$ such that $[0,1]^2\setminus K$ is the union of two disjoint open sets $T$ and $W$ with $U\subseteq T$ and $V\subseteq W$.
It seems like this is a classical result but I don't know.
Note: You may not be able to choose $K$ to be an arc because for certain open $U$ and $V$ it could have to be something like the topolgist's sine curve.