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

1. ## 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?

2. I think this looks good.

You can also show that it is not closed by observing that 0 is the limit of the sequence $\displaystyle \langle \frac{1}{n}\rangle$

3. How does that prove it?

4. Originally Posted by magus
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.

5. 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.

6. What you have done with showing 0 is a limit point of (0,1] is correct.

7. Does that mean that the part about it not containing $\displaystyle 0$ and therefore being closed is wrong?

8. Originally Posted by magus
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.

9. 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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# show that [0,1) is not an open set

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