On my lecture notes I read the following statement:
A Borel set can be either a countable union of closed sets ($F_\sigma$) or a countable intersection of open sets ($G_\delta$)
Does it mean, for example, that given a set $A \in B(X)$ (where $X$ is a topological space), $A$ is either $F_\sigma$ or $G_\delta$?
Thank you!
Let $E =([0,1]\cap \mathbb Q) \cup ([2,3] \setminus \mathbb Q))$. If this is a $G_{\delta}$ then intersecting with $(-1, 1.5)$ we see that $[0,1]\cap \mathbb Q$ is a $G_{\delta}$ but this is known to be false. Similarly, if $E$ is an $F_{\sigma}$ so is its intersection with $[1.5,4]$ which makes $[2,3] \setminus \mathbb Q$ an $F_{\sigma}$ . This is again a contradiction.
