My question is rather about terminology.
I know that $F_\sigma$ set is a countable union of closed sets, an $F_{\sigma\delta}$ is a set that can be expressed as a countable intersection of $F_\sigma$-sets etc. etc.
I have seen these notions in various papers several times and know them under "Borel hierarchy". But then, when I simply search Borel hierarchy on Wikipedia, I get $\Pi$s, $\Sigma$s, and $\Delta$s, but no mention of $F´s$ or $G´s$.
How are these notions connected? What am I missing? Thank you.
We have two different notational systems for the same thing: $F_\sigma$ is synonymous with $\Sigma^0_2$, $G_\delta$ is synonymous with $\Pi^0_2$, etc. For most purposes, once we get above the third level of the Borel hierarchy the $\Sigma/\Pi$ notation is much more convenient (this is especially true once we get into the infinite levels!), and so sources like the wiki page which present the whole hierarchy itself as opposed to focusing on a small initial segment typically use that notation.
(That said, there is one situation where the $F/G$ notation is nicer: when we drop the axiom of choice. For example, in $\mathsf{ZF}$ it is not true that $F_\sigma=F_{\sigma\sigma}$. But this is a rather technical aspect which should be ignored at first.)