Exercise On Quotient Topology Let $X = \mathbb R$ (real numbers) in the standard topology. Take the partition $X' = \{\dots, (-1 , 0] , (0 , 1] , (1 , 2] ,\dots\}$.
My question is this: 
Describe the open sets in the resulting quotient topology on $X'$?

Hint: The open subsets of $X'$ are those sets $\mathcal O\subseteq X'$ such that $\bigcup\mathcal O$ is open in $X=\Bbb R$. What can a subset of $X'$ look like in order for the union of its members to be a non-trivial open subset of $X=\Bbb R$?
