I am trying rewrite the set:
$$E_1 = (0,1] $$
as a collection of countable unions or intersections. I understand there are a ton of ways to do this, however, I just want to make sure mine is correct so I understand how to evaluate these objects. My thought was to break this into:
$$E_1=(0,1)\cup[1]=\bigcup\limits_{n=2}^{\infty}[-\frac{1}{n},1-\frac{1}{n}] \cup\bigcap\limits_{n=2}^{\infty}(1-\frac{1}{n},1+\frac{1}{n}) $$
I believe the infinite union should be equal to $(0,1)$ and the infinite intersection should be equal to $[1]$ so that their union is simply $E_1$. Is this correct?
EDIT: This was supposed to read:
$$E_1=(0,1)\cup[1]=\bigcup\limits_{n=2}^{\infty}[\frac{1}{n},1-\frac{1}{n}] \cup\bigcap\limits_{n=2}^{\infty}(1-\frac{1}{n},1+\frac{1}{n}) $$
Best Answer
The sequence of sets $\{I_n\}$ where $ I_n = [-\frac{1}{n},\frac{1}{n}]$ is decreasing, and $I_{n} \subset I_2$ (for all $n$), so its union is just $I_2$. You wont get $E_1$ like this. You are right the intersection is 1. You can write $E_1$ directly as intersection (enumerable) of open sets.
While the infinite union of open sets can be closed (i.e $ \bigcup (-n,n) = \mathbb{R}$), it is necessarily open as well, so you cannot write $E_1$ as infinite union of open sets, because $E_1$ is not open.