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October 10th, 2014, 10:53 AM   #1
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Equivalence Relation Question

Hi all!

I'm reading right now about the quotient topology and quotient spaces, and I've come across an example that uses this statement:

"On $\mathbb{R}^2$, let $\equiv$ be the equivalence relation generated by $(x, y) \equiv (-x, y)$."

I'm not familiar with this treatment of equivalence relations; I'm used to a book explicitly defining an equivalence relation in the way of "$(x, y) \equiv (a, b)$ if and only if . . ." .

Do I just treat this as "The equivalence relation on $\mathbb{R}^2$ such that every point $(x, y)$ is related to the point $(x, y)$"?

If not, how do treat this? Having a Master's, I'm well acquainted with equivalence relations, but I'm hoping I wasn't just spacing out as an undergrad on the day equivalence relations were taught in my introductory proofs class.

Last edited by matthematical; October 10th, 2014 at 10:57 AM.
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October 10th, 2014, 03:46 PM   #2
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Originally Posted by matthematical View Post
"On $\mathbb{R}^2$, let $\equiv$ be the equivalence relation generated by $(x, y) \equiv (-x, y)$."
This means that $\equiv$ is the reflexive, symmetric and transitive closure of the given relation, i.e., the smallest equivalence relation containing the given one. In this case, one only has to take the reflexive closure, I believe.
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