0
$\begingroup$

Is there somone help me to show that if this problem have positive Answer :

Problem :Can every non-discrete topological group G be algebraically gen-

erated by a nowhere dense subset ?

Thank you for any help .

$\endgroup$
5
  • 3
    $\begingroup$ Please take more effort in asking your questions. First, it is full of linguistic and spelling errors and other inconsistencies. Second, your labels are not appropriate, e.g. your question has nothing to do with proof theory (proof theory is not about asking for proofs), and little to do with normal subgroups. $\endgroup$
    – GH from MO
    Jun 9, 2015 at 1:02
  • $\begingroup$ I don't know where is the problem in my question ? $\endgroup$ Jun 9, 2015 at 1:34
  • 4
    $\begingroup$ Apparently this was open as of 2001, according to Problem 9 of researchgate.net/profile/Taras_Banakh/publication/… $\endgroup$ Jun 9, 2015 at 2:07
  • $\begingroup$ Thank you very much, i don't accross it yet .Just independent guess $\endgroup$ Jun 9, 2015 at 2:10
  • $\begingroup$ Then it's unsolved until now ? $\endgroup$ Jun 9, 2015 at 2:12

1 Answer 1

2
$\begingroup$

According to this paper on open problems in topological algebra, this is an open problem (it's listed as Problem 9 in particular).

Problem 9 (Protasov). Can every topological group be algebraically generated by a nowhere dense subset?

Let us mention that each countable topological group is algebraically generated by some closed discrete subset (see reference in [13]) while every left topological group is algebraically generated by some subset with empty interior [13].

[13] in particular is from 2001.

$\endgroup$
7
  • $\begingroup$ Yes, this seems to be the exact same paper mentioned by Nate Eldredge in his comment above. $\endgroup$
    – Todd Trimble
    Jun 9, 2015 at 2:16
  • $\begingroup$ @ToddTrimble It is exactly that. He posted it to a comment as I was typing this together. $\endgroup$ Jun 9, 2015 at 2:16
  • $\begingroup$ @Todd Trimble :ok what about the answer with nowhere not dense ? does it had a positive answer ? $\endgroup$ Jun 9, 2015 at 2:18
  • $\begingroup$ @zeraouliarafik Pardon my possible ignorance, but what do you mean by "nowhere not dense"? Is that the same as "dense"? $\endgroup$ Jun 9, 2015 at 2:22
  • $\begingroup$ yeah , for example :ensemble of Real number is nowhere but not dense ,if it's interesting I edit my quesion $\endgroup$ Jun 9, 2015 at 2:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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