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October 10th, 2014, 09:53 AM  #1 
Member Joined: Jun 2011 From: California Posts: 82 Thanks: 3 Math Focus: Topology  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 09:57 AM. 
October 10th, 2014, 02:46 PM  #2 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  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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