One of the Peano axioms state that
"For any $a \in \Bbb N : a = b, b \in \Bbb N$."
An example where transition is not closed under equality is the relation to friends. C may not be B's friend, but A may be B's friend, even though A isn't C's friend; that is : A = B =/= C.
Is there any case for any set (not limited to $\Bbb N$) where the set is not closed under equality, that is, it produces a member from another set? I would assume this to mean "open" (not in the topological definition), but correct me if I am wrong.
From Wikipedia: For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the natural numbers are closed under equality