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Topology

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Examples of Equivalence Relations in Directed Sets

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Let $X$ be a directed set. Which of the following(s) is(are) equivalence relation(s)?

A

$\sim_A$ such that $a\sim_A b$ iff $a\preceq b$ or $b\preceq a$.

B

$\sim_B$ such that $a\sim_B b$ iff there are finitely many $c$ such that $a\preceq c$ and $b\preceq c$.

C

$\sim_C$ such that $a\sim_C b$ iff there are finitely many $c$ such that $a\preceq c\preceq b$ or $b\preceq c\preceq a$.

D

$\sim_D$ such that $a\sim_D b$ iff $\exists c$, $a\preceq c$ and $b\preceq c$.