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I have a problem with this excercise. I need your help.

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$

$f(t)=t+[t]$

where $[\cdot]$ is the floor function.

Define the metric:

$$d(x, y)=|f(x)-f(y)|\quad (x,y)\in\mathbb{R}^2$$

  1. The metric space $(\mathbb{R}, d)$ is complete?

  2. Can $(\mathbb{R}, d)$ be expressed as countable union of its compact subset?

For 1. I think I have to show that a Cauchy sequence is convergent on $(\mathbb{R}, d)$ right?

For 2. I have no idea.

Help me please :)

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For 2. you could try to determine $(\mathbb R, d)$'s compact subsets ... –  martini Apr 14 '12 at 9:09
    
For 1., you could try $x_n=1-1/n$. Do you think the sequence $(x_n)$ is Cauchy? (Yes.) Does it converge? –  Did Apr 14 '12 at 9:30
    
For 2., you could try to show that $[k,k+n/(n+1)]$ is compact, for every integers $n\geqslant1$ and $k$. –  Did Apr 14 '12 at 9:32
    
Thank you, guys. –  Darkifrit Apr 14 '12 at 9:33
    
This is a typical case where you could write a full solution to your own question. –  Did Apr 14 '12 at 10:22
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1 Answer

Since $\langle\mathbb R,d\rangle$ isn’t complete, you’d be better off looking for a Cauchy sequence that doesn’t converge. For both parts it may help to prove the following facts.

  1. If $x\in\mathbb{Z}$ and $0<\epsilon\le 1$, $$B(x,\epsilon)=[x,x+\epsilon)\;.$$

  2. If $x\in\mathbb R\setminus\mathbb Z$, and $0<\epsilon\le \min\Big\{x-\lfloor x\rfloor,\lceil x\rceil-x\Big\}$, $$B(x,\epsilon)=(x-\epsilon,x+\epsilon)\;.$$

Note that this is almost what happens in $\mathbb R$ with the usual metric; the only points that behave differently are the integers. For (1) you could try to find a $d$-Cauchy sequence that would converge to some integer $n$ in the usual metric but not in this one.

For (2), show that for each integer $n$, the interval $[n,n+1)$ has exactly the same compact subsets in $\langle\mathbb R,d\rangle$ as it has in the usual topology. Is it the union of countably many of these compact subsets?

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@Didier: Thanks; fixed. I inadvertently doubled the fractional part of the other number. –  Brian M. Scott Apr 14 '12 at 10:18
    
Right. Now for item 2. –  Did Apr 14 '12 at 10:20
    
$B(.5,.75)\ne(-.25,1.25)$. –  Did Apr 15 '12 at 20:01
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