Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 3 down vote accepted

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 $[0,1]$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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