Let
R
, the real line be endowned with the discrete topology. Which of the following subsets of
R
is dense in
R

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- chestercat

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- anonymous

Q
Ritself
Qc
All singletons

- steve816

Wow, this is some advanced stuff. Just wondering, what branch of mathematics is this?

- anonymous

topology

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## More answers

- steve816

Nice, wish there was topology class in high school, did you learn about the banach-tarski paradox?

- anonymous

no

- steve816

Well, it's pretty interesting. I don't think anyone in openstudy knows anything about topology though.

- anonymous

someone does

- steve816

At least not at 3 AM in the morning lol

- anonymous

yes

- steve816

So... how hard is topology?

- anonymous

in a way very

- 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

\(U\) 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

Clearly the entire space is dense because it contains all of its own points!

- zzr0ck3r

@steve816 it really is not, and it is a fun new (new to me...)way of looking at things :)

- anonymous

so, which option is correct

- anonymous

is it R?

- zzr0ck3r

It should be very clear. Read it and tell me if you don't understand.

- 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

ok, from what you said, i its R

- anonymous

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