# 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 $(0,1]$ is not open

Since $1\in (0,1]$ there should exist some interval $1\in (a,b)$ so $a<1 because we are using the euclidean topology. Then there should also be $a<1<\frac{b+1}{2} 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. 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?

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 $\langle \frac{1}{n}\rangle$

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 $(0,1]$.
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.

5. 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.

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 $0$ and therefore being closed is wrong?

8. Originally Posted by magus
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.

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.