2
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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 $[0,1]$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.