I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to closed sets.
Let $U$ be an open connected subset of the closed unit square $[0,1]^2$ such that $U\cap \partial I^2=(0,1)\times \{0\}$. Suppose $V\subseteq U$ is open (but not necessarily connected), $V\cap \partial I^2=(1/3,2/3)\times\{0\}$, and that $V$ separates $a=(1/4,0)$ and $b=(3/4,0)$ in $U$. The last condition means that $a$ and $b$ lie in distinct connected components of $U\backslash V$.
I'd basically like to know if the component of $V$ that intersects the x-axis (call it $V_0$) limits on $[0,1]^2\backslash U$ and, alone, does the job of separating $a$ and $b$ in $U$.
Question 1: If $V_0$ is the connected component of $V$ that contains $(1/3,2/3)\times\{0\}$, must $V_0$ separate $a$ and $b$ in $U$?
Question 2: Must $\overline{V_0}$ meet $[0,1]^2\backslash U$?
