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Math Help - prove that (0,1] is neither open nor closed

  1. #1
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    prove that (0,1] is neither open nor closed

    Since I've had a marked disability to write proofs I wanted to run this by you guys to insure that it was correct before I moved on.

    First I prove that (0,1] is not open

    Since 1\in (0,1] there should exist some interval 1\in (a,b) so a<1<b because we are using the euclidean topology. Then there should also be a<1<\frac{b+1}{2}<b or \frac{b+1}{2}\in (a,b). However, \frac{b+1}{2}\not\in (0,1].

    Then we need to prove that it is not closed. To do such We prove that the compliment is not open.

    (0,1]'=(-\infty,0]\cup(1,\infty). To prove that this is not open we just need to prove that one of the members of the union is not open. Using the same strategy then on (-\infty,0] let 0\in (a,b) or a<0<b. Then find the element \frac{b}{2} and see that it is not in (-\infty,0]. Thus proving that the compliment is not open and that the set (0,1] is not open nor closed.

    Is this correct?
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  2. #2
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    I think this looks good.

    You can also show that it is not closed by observing that 0 is the limit of the sequence \langle \frac{1}{n}\rangle
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  3. #3
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    How does that prove it?
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  4. #4
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    Quote Originally Posted by magus View Post
    How does that prove it?
    You did a good job showing that 1 is not an interior point of (0,1].
    Every open set is its own interior.

    Now show that if \delta>0 the open set (-\delta,\delta) contains a point of (0,1].
    That means 0 is a limit point not in (0,1].
    So it is not closed.
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  5. #5
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    I had to look ahead to see what limit points were but I think I get it.



    x is a limit point in A if every open set contains a point from A different from x and closed sets contain all their limit points.

    So (-\delta,\delta) is our open set and it contains points different from 0 that are in (0,1], so 0 is a limit point. The interval (0,1] does not contain 0 so it is not a closed set.

    Is this right?

    Also since I've now read a bit on the chapter of limit points is say 2 not a limit point of (0,1] because there's an open interval (-3,0) that contains no points from the set (0,1] but contains [LaTeX ERROR: Convert failed] ?

    I just want to be sure I'm getting the material right and thank you all for all of your help thus far.
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  6. #6
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    What you have done with showing 0 is a limit point of (0,1] is correct.
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  7. #7
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    Does that mean that the part about it not containing 0 and therefore being closed is wrong?
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  8. #8
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    Quote Originally Posted by magus View Post
    Does that mean that the part about it not containing 0 and therefore being closed is wrong?
    Quite the opposite. It shows that it is not closed.
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  9. #9
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    Opps I mixed up what I was trying to prove. So it proves that it's not closed and above I proved that it's not open. Okies. I think I should sleep now. Thank you.
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