the above notation, we have $\Omega\subseteq\{(t,x)\mid t\in I,\, a(t)<x<b(t)\}$ if the set $\Omega$ is open.

\[def:XSimpleSet\] We call a subset $U$ of the $(t,x)$-plane an *$x$-simple set* if the intersection of $U$ by any line $t=t_0\in\mathbb R$ is an open interval within this line or the empty set.

Equivalently, a subset $U$ of the $(t,x)$-plane is called an $x$-simple set if there exist a subset $I$ of the $t$-axis and functions $a,b\colon I\to\mathbb R\cup\{-\infty,+\infty\}$ with $a(t)<b(t)$ for any $t\in I$ such that $$\begin{gathered}
\label{eq:EquivDefOfXSimpleSet}
U=\{(t,x)\mid t\in I,\, a(t)<x<b(t)\}.\end{gathered}$$ Then $I={\mathop{\rm pr}\nolimits}_tU$, $a={\rm lb}_U$ and $b={\rm ub}_U$. Note that this definition of $x$-simple set is similar to but in fact different from and, in certain sense, more general than the one used in elementary calculus (cf., e.g., ).

\[lem:OnOpenX-SimpleRegion\] An $x$-simple subset $\Omega$ of the $(t,x)$-plane is open if and only if its projection $I:={\mathop{\rm pr}\nolimits}_t\Omega$ is an open subset of $\mathbb R$ and its lower and upper bounds, $a:={\rm lb}_\Omega$ and $b:={\rm ub}_\Omega$, are upper and lower semi-continuous functions on $I$, respectively. Moreover, in this case there exists a smooth function $\theta\colon I\to\mathbb R$ such that $a(t)<\theta(t)<b(t)$ for any $t\in I$.

![Objects related to $x$-simplicity[]{data-label="fig:ObjectRelatedToXSimplicity"}](Ris1){width=".6\linewidth"}

The necessity of the first 