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For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap Q$ a clopen subset of the metric space $\mathbb{Q}$?

My answer is : $[a,b]\cap\mathbb{Q}$ is a clopen subset iff $a,b \in (\mathbb{R}\backslash \mathbb{Q})$, since if $a,b \in \mathbb{Q}$ then $[a,b]\cap \mathbb{Q}$ won't be open.

I got this wrong on a p set. Can someone correct what I have done wrong?

user98727
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2 Answers2

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Well, one thing is that you didn't include the case where $b\lt a$, since in this case the interval is empty, which is both open and closed.

But also, the logic of your argument isn't quite right, since if $a$ and $b$ are not both irrational, it doesn't mean that they are both rational, but rather it means only that one of them is rational. So you've got to argue that if one of $a$ or $b$ is rational, then the interval is not clopen. But I think it will not be difficult for you to fix that issue.

JDH
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Your answer is correct if you consider the expression $[a,b]$ ill-formed when $a>b$, but there’s a small flaw in your justification: you’ve said that if $a$ and $b$ are both rational, then $[a,b]\cap\Bbb Q$ is not open (in $\Bbb Q$), which is correct, but to justify your answer you also have to point out that $[a,b]\cap\Bbb Q$ isn’t open in $\Bbb Q$ if even one of $a$ and $b$ is rationa.

If you allow intervals $[a,b]$ with $a>b$, then you must include them as clopen irrespective of whether $a$ or $b$ is rational, since they’re all empty.

Brian M. Scott
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  • The instructor gave me 1/5 points, is that too harsh? – user98727 Oct 10 '13 at 02:04
  • @user98727: Let me ask first: does your class allow intervals like $[2,1]=\varnothing$, or do you take $[2,1]$ to be meaningless, since $2>1$? – Brian M. Scott Oct 10 '13 at 02:09
  • Probably meaningless – user98727 Oct 10 '13 at 02:14
  • Another issue is whether your instructor asked the question literally as you did, that is, for intervals, or whether he asked for all pairs of reals numbers $a$ and $b$ for which the interval is clopen. If he or she really asked just for intervals, then you can argue that your answerr is fully right, since the intervals with $b\lt a$ are empty, and hence equal to the interval $[\sqrt{2},\sqrt{2}]\cap\mathbb{Q}$, which is among your answer. So you've given a complete list of all the intervals having the clopen property. (But this is apart from the other logical flaw in your argument.) – JDH Oct 10 '13 at 02:15
  • @user98727: In that case I’d probably have awarded $3$, maybe even $4$, out of $5$, assuming that you accurately quoted or paraphrased your actual answer. $1/5$ seems a bit harsh to me, but can’t really judge it in isolation: it might be consistent with your instructor’s overall marking habits. How I’d judge it also depends on how the instructor interprets the final totals. If the best students are expected to get $95$%; it’s pretty harsh. If they’re expected to get around $80$%, as was usually the case for me, it’s less harsh (though still on the harsh side in my view). And so on. – Brian M. Scott Oct 10 '13 at 02:22