we denote by $C$. According to the Jordan curve theorem, this curve divides the $(t,x)$-plane into the (bounded) interior $\Gamma$ and the (unbounded) exterior $\tilde\Gamma$. Up to reflections in $t$, we can assume that there exists a neighborhood $U$ of the point $(t_0,x_1)$ such that $U\cap\Gamma$ and $U\cap\tilde\Gamma$ contain only points with negative and positive values of $t-t_0$, respectively. Set $$\tilde t_0=\inf\{t\in\mathbb R\mid \exists\ x\in\mathbb R\colon (t,x)\in(C\cup\Gamma)\setminus\Omega\}.$$

![Existence of a rectangle with a punctured boundary in an open connected non-$x$-simple set[]{data-label="fig:OnOpenConnectedNonX-SimpleSets"}](Ris4){width="1.\linewidth"}

If $\tilde t_0=t_0$, then we set $\tilde x_1=x_1$, $\tilde x_2=x_2$ and $\varepsilon=\frac12\mathop{\rm dist}\big(\{\tilde t_0\}\times[\tilde x_1,\tilde x_2],\gamma([0,1])\big).$ (The distance is measured between disjoint compact sets and thus $\varepsilon>0$.)

=-1 Otherwise, the compactness of $(C\cup\Gamma)\setminus\Omega$ implies that there exists $\tilde x_0\in\mathbb R$ with $(\tilde t_0,\tilde x_0)\in\Gamma\setminus\Omega$. Since the set $\Gamma$ is bounded, the line $t=\tilde t_0$ intersects the curve $C$ in at least one point with $x$-coordinate less than $\tilde x_0$ and in at least one point with $x$-coordinate greater than $\tilde x_0$. Therefore the values $$\begin{gathered}
\hat x_1=\sup\big\{x\in\mathbb R\mid (\tilde t_0,x)\in\gamma([0,1]),\, x<\tilde x_0\big\}, \quad
\tilde x_1=\inf\big\{x\in\mathbb R\mid (\tilde t_0,x)\notin\Omega,\, x>\hat x_1\big\},
\\
\hat x_2=\inf\big\{x\in\mathbb R\mid (\tilde t_0,x)\in\gamma([0,1]),\, x>\tilde x_0\big\}, \quad
\tilde x_2=\sup\big\{x\in\mathbb R\mid (\tilde t_0,x)\notin\Omega,\, x<\hat x_2\big\}\end{gathered}$$ are well defined 