anonymous
  • anonymous
Let R , the real line be endowned with the discrete topology. Which of the following subsets of R is dense in R
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
Q Ritself Qc All singletons
steve816
  • steve816
Wow, this is some advanced stuff. Just wondering, what branch of mathematics is this?
anonymous
  • anonymous
topology

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More answers

steve816
  • steve816
Nice, wish there was topology class in high school, did you learn about the banach-tarski paradox?
anonymous
  • anonymous
no
steve816
  • steve816
Well, it's pretty interesting. I don't think anyone in openstudy knows anything about topology though.
anonymous
  • anonymous
someone does
steve816
  • steve816
At least not at 3 AM in the morning lol
anonymous
  • anonymous
yes
steve816
  • steve816
So... how hard is topology?
anonymous
  • anonymous
in a way very
zzr0ck3r
  • zzr0ck3r
All sets are open in the discrete topology. Consider any any set \(U\ne X\). Then \(U^C\) is an open set disjoint from \(U\). This shows that the only dense set is the whole space.
zzr0ck3r
  • zzr0ck3r
\(U\) is dense means that for any \(x\) in the space \(X=\mathbb{R} _D\), either \(x\) is in \(U\) or \(x\) is a limit point of \(U\). So we take an element \(x\) that is not in \(U\) and see if it is a limit point of \(U\). If \(x\) is a limit point of \(U\) then every open neighborhood around \(x\) intersects \(U\). But since every set is open, due to the topology, we have that \(U^C\) itself is open and contains \(x\) (remember \(x\notin U\)) and of course \(U\cap U^C=\emptyset\). So \(U\) contains all of its limit points. Since \(U\) was not all of \(\mathbb{R}_D\) we have that the only dense set is the entire space.
zzr0ck3r
  • zzr0ck3r
Clearly the entire space is dense because it contains all of its own points!
zzr0ck3r
  • zzr0ck3r
@steve816 it really is not, and it is a fun new (new to me...)way of looking at things :)
anonymous
  • anonymous
so, which option is correct
anonymous
  • anonymous
is it R?
zzr0ck3r
  • zzr0ck3r
It should be very clear. Read it and tell me if you don't understand.
zzr0ck3r
  • zzr0ck3r
I want you to at least be sure yourself. I did not write all of that out so that you could ask which option it is, click a button, and move on.
anonymous
  • anonymous
ok, from what you said, i its R
anonymous
  • anonymous
@zzr0ck3r

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