#Real analysis - Density?

14 messages · Page 1 of 1 (latest)

cloud wedge
#

Can someone please explain what it means for a set to be dense?

More specifically, what is the difference/relationship between density and compactness?

I'm trying to teach myself some material, but I'm having a hard time understanding, so I thought I'd ask! Thank you so much in advance!

ashen archBOT
#
  1. Do not ping the Moderators, unless someone is breaking the rules.
  2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
  3. Do not ping other members randomly for help.
  4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  5. Wait patiently for a helper to come along.
  6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

minor gulch
#

Density and compactness are quite different concepts. Some good examples to consider:\begin{itemize}\item$\mathbb{Q}$, the set of rational numbers, is dense in $\mathbb{R}$.\\item$[0,1]$, the closed unit interval, is compact.\end{itemize}

supple flameBOT
#

Pear Category Theorem

326954635287855104.png
minor gulch
#

Clearly, Q and [0,1] behave quite differently. You should think of "dense" as "being like Q", and "compact" as "being like [0,1]".

#

I think I should specify further. The density of $\mathbb{Q}$ basically means that you can approximate any real number by a rational number, arbitrarily well. For example, $\pi$ is irrational. But you can get rational numbers progressively closer and closer to the real value of $\pi$. Consider $3,3.1,3.14,3.141,3.1415,3.14159,\dots$

supple flameBOT
#

Pear Category Theorem

326954635287855104.png
minor gulch
#

Each term in that sequence is by itself rational. But, they together approximate pi, which is irrational.

dense silo
#

Or more precisely.

#

Take a set A and a subset B and some ordering relation represented by <. Then B is dense in A if and only if, for all x and y in A such that x < y, there exists z in B such that x < z < y.

#

Q is dense in Q and in R.

stoic grailBOT
#

@cloud wedge

:HelpIcon:| Help Reminder

Hello finchh_, this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.