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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 :)

  • 0
    This is a typical case where you could write a full solution to your own question.2012-04-14

1 Answers 1

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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?

  • 0
    $B(.5,.75)\ne(-.25,1.25)$.2012-04-15