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 $X'$?

For $n\in\Bbb Z$ let $p_n=(n-1,n]$, so that $X'=\{p_n:n\in\Bbb Z\}$. Let $q:X\to X'$ be the quotient map: $q(x)=p_n$ if and only if $n-1<x\le n$, i.e., if and only if $\lceil x\rceil=n$. By definition a set $U\subseteq X'$ is open in $X'$ if and only if $q^{-1}[U]$ is open in $X$. 
Now $q^{-1}[U]$ is always a union of intervals of the form $(n-1,n]$; under what circumstances is such a union open in $X$? Suppose, for instance, that $p_1\in U$, so that one of the intervals is $(0,1]$. In order for $q^{-1}[U]$ to be open in $X$, it must contain an ordinary open interval around $1$, and the only way to get that is to have $p_2\in U$, so that $(1,2]\subseteq q^{-1}[U]$. Now we know that $q^{-1}[U]\supseteq(0,2]$, so in order for it to be open, it must contain an open interval around $2$. The only way to get that is to have $p_3\in U$, so that $(2,3]\subseteq q^{-1}[U]$. Keep going: by induction you can show that if $p_1\in U$, and $U$ is open in $X'$, then $p_k\in U$ for each $k\ge 1$, and $q^{-1}[U]$ contains all of the positive reals. In fact, we can now see that $\{p_n:n\ge 1\}$ is open in $X'$: its inverse under $q$ is just the positive reals.
Now try to generalize from that example to find the rest of the open sets in $X'$.
