claim follows from Lemma \[lem:OnOpenSetLB&UB\]. Let us prove its sufficiency. Suppose that the set $I$ is open and the functions $a$ and $b$ are upper and lower semi-continuous, respectively. Fix an arbitrary $(t_0,x_0)\in\Omega$. Then $a(t_0)<x_0<b(t_0)$ and thus $\varepsilon:=\frac12\min\{x_0-a(t_0),b(t_0)-x_0,1\}>0$. Hence there exists a neighborhood $V$ of $t_0$ in $I$ such that for all $t\in V$, $a(t)<x_0-\varepsilon$ and $b(t)>x_0+\varepsilon$, i.e., the neighborhood $V\times(x_0-\varepsilon,x_0+\varepsilon)$ of $t_0$ is contained in $\Omega$. Therefore, the set $\Omega$ is open.

Supposing now that $\Omega$ is $x$-simple, we can cover the open set $I$ by a family of open subsets $(V_j)_{j\in J}$, where $J$ is some index set, such that for any $j\in J$ there exists $c_j\in\mathbb R$ with $a(t)<c_j<b(t)$ for any $t\in V_j$. Let $(\chi^j)_{j\in J}$ be a ${\rm C}^\infty$-partition of unity that is subordinate to $(V_j)_{j\in J}$, i.e., $({\mathop{\rm supp}\nolimits}\chi^j)_{j\in J}$ is locally finite and ${\mathop{\rm supp}\nolimits}\chi^j\subseteq V_j$ for any $j\in J$ [@MadsenTornehave1997 Theorem A.1]. Then the function $\theta:=\sum_{j\in J}\chi^jc_j$ belongs to ${\rm C}^\infty(I)$, and for any $t\in I$ we obtain $$a(t)=\sum_{j\in J}\chi^j(t)a(t)<\sum_{j\in J}\chi^j(t)c_j=\theta(t)<\sum_{j\in J}\chi^j(t)b(t)=b(t).$$ (Here non-strict inequalities are clear for any $j\in J$, but also for any $t\in I$ there exists $j_0\in J$ with $\chi^{j_0}(t)>0$, and for this term we have $\chi^{j_0}(t)a(t)<\chi^{j_0}(t)c_j<\chi^{j_0}(t)b(t)$, which implies that 