## An equivalence relation on the power set of the plane.

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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.

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## 1 Answer

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 $[0,1]$.

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