Let $X$ be a Polish space. We know that $A\subset X$ is Polish iff it is a $G_\delta$ subset. So, since every open and every closed subset is $G_\delta$, it is in particular Polish.
Now, we let $A\subset X$ be an open subset and obtained two Polish spaces $A$ and $X\setminus A$. My question is as follows: If $B\subset X\setminus A$ open, is it Polish again and furthermore, is $(X\setminus A)\setminus B$ also Polish?
Translated in $X=\mathbb{R}$, choose $A=(0,1)$, so $X\setminus A=(-\infty, 0]\cup[1, \infty)$, then $B=[1,2)$ is again open in $X\setminus A$ and thus Polish as well as $(X\setminus A)\setminus B =(-\infty, 0]\cup[2, \infty)$.
But does this hold in general, that is, for $A_1, A_2\subset X$ open, does it always hold that $A_1\setminus A_2$ is Polish?
I have tried to show that it is a $G_\delta$ set but I was not so successful with it
The answer is yes. Just mentally replace $X\setminus A$ with $X$: once you’ve established that $X\setminus A$ is Polish, it remains true that any open subset of it is a Polish subspace.