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Thread: 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 $\displaystyle (0,1]$ is not open

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

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

    $\displaystyle (0,1]'=(-\infty,0]\cup(1,\infty)$. 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 $\displaystyle (-\infty,0]$ let $\displaystyle 0\in (a,b)$ or $\displaystyle a<0<b$. Then find the element $\displaystyle \frac{b}{2}$ and see that it is not in $\displaystyle (-\infty,0]$. Thus proving that the compliment is not open and that the set $\displaystyle (0,1]$ 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 $\displaystyle \langle \frac{1}{n}\rangle$
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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 $\displaystyle (0,1]$.
    Every open set is its own interior.

    Now show that if $\displaystyle \delta>0$ the open set $\displaystyle (-\delta,\delta)$ contains a point of $\displaystyle (0,1]$.
    That means 0 is a limit point not in $\displaystyle (0,1]$.
    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.



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

    So $\displaystyle (-\delta,\delta)$ is our open set and it contains points different from $\displaystyle 0$ that are in $\displaystyle (0,1]$, so $\displaystyle 0$ is a limit point. The interval $\displaystyle (0,1]$ does not contain $\displaystyle 0$ 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 $\displaystyle 2$ not a limit point of $\displaystyle (0,1]$ because there's an open interval $\displaystyle (-3,0)$ that contains no points from the set $\displaystyle (0,1]$ but contains $\displaystyle 2$?

    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 $\displaystyle 0$ 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 $\displaystyle 0$ 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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