An equivalence relation on the power set of the plane.

Question

Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H_R =$ {$ Rb|b\in\mathbb{R}$} where $Rb=${$ a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections" of $R$, $V_R =${$ aR|a\in\mathbb{R}$} where $aR=${$ b\in\mathbb{R} | (a,b)\in R$}. Now define the equivalence relation on $\wp (\mathbb{R^2})$ such that $R \sim S$ if, and only if, $H_R=H_S$ and $V_R=V_S$.

1. Do you have any reference to this equivalence relation or a similar one?
2. What connections does it have to topology?
3. As an example, ¿can you describe the equivalence class of a disk?

Of course this can be generalized to any set of binary relations, but I want to understand it in the case of the plane.

Answer 1

The equivalence class of the closed unit disk $ \{(x,y): x^2 + y^2 \le 1 \}$ consists of sets $S = \{(x,y) \in [-1,1] \times [-1,1]: |y| \le f(|x|)\}$ where $f$ is a decreasing homeomorphism from $[0,1]$ onto $[0,1]$.