Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Hi everyone: Is a null set of $\mathbb{R}^n$, $(n>0)$, necessarily closed? Give a counter example. Thanks for your reply.

share|improve this question

closed as off-topic by Jonas Meyer, Rolf Hoyer, Yes, John, Claude Leibovici May 4 at 6:19

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jonas Meyer, Rolf Hoyer, Yes, John, Claude Leibovici
If this question can be reworded to fit the rules in the help center, please edit the question.

    
I up-voted this question and its total is $0$, which means (1) someone down-voted it and (2) Why are those who've answered this and who've voted on the answers neglecting to up-vote it? –  Michael Hardy Jul 5 '14 at 18:30
    
Dear @MichaelHardy : the solutions have merit, but the question is only a PSQ, so this distribution seems natural. Regards –  rschwieb Jul 6 '14 at 0:08
    
@rschwieb : I'd rather not spend vast amounts of effort trying to decrypt your comment, especially when they would probably not succeed. Can you translate it into English? This web page doesn't help: acronyms.thefreedictionary.com/PSQ –  Michael Hardy Jul 6 '14 at 15:09
    
Dear @MichaelHardy : Sorry, I thought you were aware of this abbreviation, which we use regularly in meta. From the context, I believe you can deduce the meaning of the sentence is "The solutions have merit, but the question does not." Hope this makes things clear. Regards –  rschwieb Jul 6 '14 at 16:51
    
Could someone explain why my second answer below was down-voted? –  Michael Hardy Jul 6 '14 at 18:22

4 Answers 4

up vote 7 down vote accepted

Any countable set is a null set, there are many nonclosed countable sets. eg $\mathbb{Q}^n$

share|improve this answer

The set $\{\frac1n\}_{n\geq 1}$ is a nice countable (and therefore zero-measure) set in $\mathbb{R}$ that isn't closed, because it doesn't contain its limit point $0$.

share|improve this answer

$$ \left\{ \frac 1 n : n=1,2,3,\ldots \right\} $$

share|improve this answer

The set of members of the Cantor set that are not endpoints of the deleted intervals is an uncountable set of measure $0$ that is not closed.

share|improve this answer
1  
Is there a reason why this was down-voted? –  Michael Hardy Jul 6 '14 at 15:08
    
Dunno. I thought it was a good answer. –  G Tony Jacobs Jul 6 '14 at 16:02

Not the answer you're looking for? Browse other questions tagged or ask your own question.