Find all open sets of X, where X is the quotient space of $\mathbb{R}$ recieved from the following equivalence relation:
$$x \sim y \iff \exists n\in \mathbb{Z} : n<x,y\leq n+1$$
Is it true that all the open sets of X are from the form $I_n = (n, \infty)$? ( and obviously the empty set and the space X itself )