Determinate the quotient topology

Question

I was trying to find the quotient topolgy for the next example:

Let R be the real numbers with the usual topology ($\tau$) and define the relationship $\mathcal{R}$ over R as follows,

a $\mathcal{R}$ b if and only if a-b $\in$ Z (that is the difference is an integer)

I know that we the define the topolgy over R$/_{\mathcal{R}}$ as $\ \\ $ $\tau/_{\mathcal{R}}:=\{V \subseteq$ R$/_{\mathcal{R}}$| $\pi(V) \in \tau\}$ where

$\pi :$R$\rightarrow$R$/_{\mathcal{R}} \ : x \rightarrow [x]$.

I've trying to find the topology generated by this relation, but I'm having some problems trying to find the inverse image of the subsets of R$/_{\mathcal{R}}$ to determinate how are the open sets in this quotient space.

I'd really appreciate any advise or hint you can give me. Thanks so much for the help.

Answer 1

HINT: You can write any real number $x$ as $x=\lfloor x\rfloor+\{x\}$, where $\lfloor x\rfloor$ is the floor of $x$ (i.e., the greatest integer less than or equal to $x$), and $\{x\}$ is the fractional part of $x$, equal to $x-\lfloor x\rfloor$. For example, $\lfloor 3.2\rfloor=3$ and $\{3.2\}=0.2$, while $\lfloor -3.2\rfloor=-4$ and $\{-3.2\}=0.8$.

• Show that $x\mathrel{\mathscr{R}}y$ if and only if $\{x\}=\{y\}$.

Then use the fact that the set of possible values of $\{x\}$ is $[0,1)$.