certain configuration that will allow us to construct differential operators on $\Omega$ with ‘problematic’ behavior.

\[lem:OnOpenConnectedNonX-SimpleSets\]=-1 For any open connected non-$x$-simple subset $\Omega$ of the $(t,x)$-plane, there exist $\tilde t_0,\varepsilon,\tilde x_1,\tilde x_2\in\mathbb R$ with $\varepsilon>0$, and $\tilde x_1\leqslant\tilde x_2$ such that, up to reflections in $t$, the set $\Omega$ does not intersect a closed subset $\Upsilon$ of the line segment $\{\tilde t_0\}\times[\tilde x_1,\tilde x_2]$ with $(\tilde t_0,\tilde x_1),(\tilde t_0,\tilde x_2)\in\Upsilon$ and contains the subset $$\begin{gathered}
\label{eq:SpecialRectangleDomain}
[\tilde t_0-\varepsilon,\tilde t_0]\times[\tilde x_1-\varepsilon,\tilde x_2+\varepsilon]\setminus\Upsilon.\end{gathered}$$

Since the set $\Omega$ is not $x$-simple, there exists $t_0\in{\mathop{\rm pr}\nolimits}_t\Omega$ such that $\Omega_{t_0}$ is not connected, i.e., for some $x_0,x_1,x_2,x_3\in\mathbb R$ with $x_0<x_1\leqslant x_2<x_3$ we have $[x_0,x_1),(x_2,x_3]\subset\Omega_{t_0}$ and $x_1,x_2\notin\Omega_{t_0}$; see Figure \[fig:OnOpenConnectedNonX-SimpleSets\]. Since the set $\Omega$ is connected, there exists a (continuous) path $\gamma\colon[0,1]\to\Omega$ with $\gamma(0)=(t_0,x_0)$ and $\gamma(1)=(t_0,x_3)$. Without loss of generality, we can assume the map $\gamma$ injective.[^1] Let $$\tau_0=\sup\big\{\tau\in[0,1]\mid\gamma(\tau)\in\{t_0\}\times[x_0,x_1)\big\},\quad
\tau_1=\inf\big\{\tau\in[0,1]\mid\gamma(\tau)\in\{t_0\}\times(x_1,x_3]\big\}.$$ Replacing $\gamma$ by its subpath $\gamma\big|_{[\tau_0,\tau_1]}$, $(t_0,x_0)$ by $\gamma(\tau_0)$, $(t_0,x_3)$ by $\gamma(\tau_1)$ and $x_2$ by the supremum of the relative complement of $\Omega_{t_0}$ in the new interval $(x_0,x_3)$, we can also assume that $(t_0,x_0)$ and $(t_0,x_3)$ are the only common points of $\gamma([0,1])$ with $\{t_0\}\times[x_0,x_3]$. We complete $\gamma([0,1])$ by $\{t_0\}\times(x_0,x_3)$ to a simple closed curve, which 