(a) Let X be a topological space. Prove that the set $Homeo(X)$ of homeomorphisms $f:X \to X$ becomes a group when endowed with the binary operation $f \circ g$ .
(b) Let $G$ be a subgroup of $Homeo(X)$. Prove that the relation "$xR_G y \iff \exists g \in G$ such that $g(x)=y$" is an equivalence relation.
I have done part (a) but I am unsure what part (b) is really even saying, I don't understand what the proposed equivalence relation means so am not sure how to prove it is an equivalence relation.