Product space of an open interval and a variable open interval

Question

I would like to construct an open subset of $\mathbb{R}^2$ in the following way. First I construct an open interval in $\mathbb{R}$ given by $X=(a,b)$. For every point $x\in X$, I define an open interval in $\mathbb{R}$ given by $Y[x]=(f[x], g[x])$ where $f$ and $g$ are smooth functions of $x$. An open subset of $\mathbb{R}^2$ is constructed as $C=\{(x,y)|x\in X, y\in Y[x]\}$. It is clear that $C$ is not a direct product space. I am wondering if there is any special name for this space?

Answer 1

Another way to write $C$ is $\{(x, y) \mid a < x < b, f(x) < y < g(x)\}$. It is the region between the graphs of $f(x)$ and $g(x)$ for $x \in (a, b)$. This assumes $f(x) < g(x)$ (otherwise, the interval $(f(x), g(x))$ is empty).

In the following image (taken from here), $C$ is the yellow region. In this example, $a = 1$, $b = 3$, $f(x) = x^2 + 3x + 1$, and $g(x) = x^2+2x+11$.

It is true that $C$ is not a direct product space in general, but it can be a product space (precisely when $f$ and $g$ are constant).