Borel Hierarchy with the usual topology in $\mathbb{R}$
In a similar proof with $[0,1)$, I rewrite this set as $\{0\}\cup (0,1)$ and then said that this is the union of a closed set with an open set, thus it was $\Delta _2^0$. Now with a set like $\displaystyle \bigcup \limits _{n\in \mathbb{N}}\left [\frac{1}{n+2},\frac{1}{n+1}\right )$, I am tempted to say is $\Delta _2^0$, but am not sure how to prove it.
Maybe my reasoning with $[0,1)$ is wrong?