If the cardinality of the set of real numbers in the interval $[0, 1]$ is $c$, and the cardinality of another set of real numbers, say those in the interval $[0, n]$ where $n>1$ is also $c$, then is it correct to say that the first interval must be more dense than the second?

  • $\begingroup$ No. The two sets are in one-to-one correspondence. $\endgroup$ – Matthew Leingang Nov 21 '17 at 14:12
  • $\begingroup$ I would agree with Matthew. What is your definition of “density”? $\endgroup$ – Julien Nov 21 '17 at 14:13
  • $\begingroup$ This question is not really well-defined because density is a property of a subset in a topological space. The question is about a set being dense as a property of a set, without saying in which space we are viewing the set. $\endgroup$ – DWe1 Nov 21 '17 at 14:16
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    $\begingroup$ @AsafKaragila I fail to see how this is a duplicate. The asker explicitly stated that they already know that there are equally many elements in the two sets. This is not about counting the elements. I do agree that the question might need some work, but closing it as a duplicate is wrong in my opinion. $\endgroup$ – Arthur Nov 21 '17 at 14:21
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    $\begingroup$ @Arthur: You are correct, and this is definitely my fault. But it seems that the OP is not entirely sure what they mean when they say "dense". Which may or may not imply it has to do with cardinality rather than any topological or order-related definition. So until this is clarified to a meaningful question, I'm reluctant to reopen this again. I mean, even your first instinct was to post an answer which talked about cardinality. $\endgroup$ – Asaf Karagila Nov 21 '17 at 14:29