Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation.
This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don't really see how to do this for this question. Any suggestions or hints are appreciated!
Best Answer
HINT: Perhaps the easiest way to prove that this is an equivalence relation is to show that the connected components of a space $X$ partition $X$: they are pairwise disjoint, and their union is $X$. It will then follow immediately from the relationship between equivalence relations and partitions that $x\sim y$ is an equivalence relation: it’s the equivalence relation induced on $X$ by the partition of $X$ into connected components.
If you need a review of the relationship between equivalence relations and partitions, I discussed it at some length in this answer.