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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

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  1. anonymous
    • one year ago
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    Q Ritself Qc All singletons

  2. steve816
    • one year ago
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    Wow, this is some advanced stuff. Just wondering, what branch of mathematics is this?

  3. anonymous
    • one year ago
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    topology

  4. steve816
    • one year ago
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    Nice, wish there was topology class in high school, did you learn about the banach-tarski paradox?

  5. anonymous
    • one year ago
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    no

  6. steve816
    • one year ago
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    Well, it's pretty interesting. I don't think anyone in openstudy knows anything about topology though.

  7. anonymous
    • one year ago
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    someone does

  8. steve816
    • one year ago
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    At least not at 3 AM in the morning lol

  9. anonymous
    • one year ago
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    yes

  10. steve816
    • one year ago
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    So... how hard is topology?

  11. anonymous
    • one year ago
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    in a way very

  12. zzr0ck3r
    • one year ago
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    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.

  13. zzr0ck3r
    • one year ago
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    \(U\) 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.

  14. zzr0ck3r
    • one year ago
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    Clearly the entire space is dense because it contains all of its own points!

  15. zzr0ck3r
    • one year ago
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    @steve816 it really is not, and it is a fun new (new to me...)way of looking at things :)

  16. anonymous
    • one year ago
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    so, which option is correct

  17. anonymous
    • one year ago
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    is it R?

  18. zzr0ck3r
    • one year ago
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    It should be very clear. Read it and tell me if you don't understand.

  19. zzr0ck3r
    • one year ago
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    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.

  20. anonymous
    • one year ago
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    ok, from what you said, i its R

  21. anonymous
    • one year ago
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    @zzr0ck3r

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