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.



An equivalence relation on the power set of the plane.
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 $[0,1]$. 


