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

This Question is Closed

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Q Ritself Qc All singletons

steve816
 one year ago
Best ResponseYou've already chosen the best response.0Wow, this is some advanced stuff. Just wondering, what branch of mathematics is this?

steve816
 one year ago
Best ResponseYou've already chosen the best response.0Nice, wish there was topology class in high school, did you learn about the banachtarski paradox?

steve816
 one year ago
Best ResponseYou've already chosen the best response.0Well, it's pretty interesting. I don't think anyone in openstudy knows anything about topology though.

steve816
 one year ago
Best ResponseYou've already chosen the best response.0At least not at 3 AM in the morning lol

steve816
 one year ago
Best ResponseYou've already chosen the best response.0So... how hard is topology?

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.0All 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
 one year ago
Best ResponseYou've already chosen the best response.0\(U\) 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
 one year ago
Best ResponseYou've already chosen the best response.0Clearly the entire space is dense because it contains all of its own points!

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.0@steve816 it really is not, and it is a fun new (new to me...)way of looking at things :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so, which option is correct

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.0It should be very clear. Read it and tell me if you don't understand.

zzr0ck3r
 one year ago
Best ResponseYou've already chosen the best response.0I 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
 one year ago
Best ResponseYou've already chosen the best response.0ok, from what you said, i its R
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