Let $X:=[0,1]$ and $V:= X \cap \mathbb{Q}= \{v_1,v_2,...\}$. For $n,k \ge1$ set $I_{n,k}:= X \cap (v_n-2^{-(n+k)},v_n+2^{-(n+k)}) $. Is it true that $$ \bigcup_{n\ge1} \bigcap_{k\ge1} I_{n,k} = \bigcap_{k\ge1} \bigcup_{n\ge1} I_{n,k} \ \ \ ?$$
It is pretty straightforward to prove that the left side is equal to $V$ (please correct me if I'm wrong). One can also show that the left side is included in the right side. But I am still not sure if this inclusion is real or if there is in fact equality between the two expressions. My main problem is, that I cannot make proper sense of the right side. Thanks in advance for any idea!
You're wrong, the left side is not included in the right side.
Since $2^{-(n+k)}$ is decreasing in $k$, $$\bigcup_{k\ge 1} I_{n,k} = I_{n,1} = X \cap (v_n - 2^{-n-1},v_n + 2^{-n-1})$$ Take some $v_n$ that is near $0$ and $v_m$ that is near $1$ and you'll have $I_{n,1} \cap I_{m,1} = \emptyset$. So the right side is empty.
Or did you mean $\bigcap_{k \ge 1} \bigcup_{n \ge 1}$?