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?