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Recall that the integer part (or integral part) of a real number x is the unique integer $n \in \mathbb Z$ such that $n \le x \lt n+1$. We denote it by $I(x)$.

On $\mathbb R$ we define the relation $xRy \iff I(x) = I(y)$.

(a) Prove that R is an equivalence relation.

(b) Let $p:\mathbb R \to \mathbb R /R$ be the quotient map, let $\mathbb R /R$ be endowed with the quotient topology, and let U be an open set in $\mathbb R /R$. Prove that if $n \in \mathbb Z$ is such that $p(n) \in U$ then $p(n-1) \in U$.

(c) Deduce that the open sets in $\mathbb R /R$ are $\emptyset,\mathbb R /R$ and the image sets $p(-\infty,n]$, where $n \in \mathbb Z$.

(d) Consider the map $I:\mathbb R \to \mathbb Z, x \mapsto I(x)$. Is the map I continuous (when $\mathbb Z$ is endowed with the subspace topology)?

I have proved that R is an equivalence relation and am stuck on part (b). I don't really see how it that is true... and being unable to do part (b) is making it hard to do (c) since they seem to build on each other. Any help with (b) would be very much appreciated.

For (b), since U is an open set in $\mathbb R /R$ then $p^{-1}(U)$ is an open set in $\mathbb R$ because of the definition of quotient topology. So if $p(n) \in U$, then $n \in p^{-1}(U)$? which is open but I don't know how to show something about $n-1$ ?

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Hint: For (b) note that $U \subseteq \mathbb{R}/R$ is open iff $p^{-1} [ U ] \subseteq R$ is open. Also, given $x \in \mathbb{R}$, if $p(x) \in A \subseteq \mathbb{R} / R$, then $[ I(x) , I(x)+1 ) = p^{-1} [ \{ p(x) \} ] \subseteq p^{-1} [ A ]$. Thus, if $p(n) \in U$ this means that $[ n , n+1 ) \subseteq p^{-1} [ U ]$, and so $p^{-1} [ U ]$ is an open neighbourhood of $n$ in $\mathbb{R}$.

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I follow what youre saying but still can't see how that shows that p(n-1) is in U...maybe its obvious and I just don't see it? –  Melissa Mar 27 '13 at 13:26
@Melissa: Here is the next step: Since $p^{-1} [ U ]$ is an open neighbourhood of $n$, this means that there is an $\epsilon > 0$ such that $( n - \epsilon , n + \epsilon ) \subseteq p^{-1} [ U ]$. In particular $p^{-1} [U]$ will have nonempty intersection with $[ n-1 , n )$. (I hope this is enough for you to conclude that $p(n-1) \in U$.) –  Arthur Fischer Mar 27 '13 at 13:29
Ah yes! Thank you very much!! –  Melissa Mar 27 '13 at 13:37
@Melissa: You're welcome! –  Arthur Fischer Mar 27 '13 at 13:40

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