Nisnevich topology inspired by Adeles I'm quite a  newbe in the field of motives & A1 homotopy theory,
so please forgive me if the question is too elementary:
In the intro from  wikipedia on Nisnevish topology
is remarked that it's founder Yevsey Nisnevich was originally motivated
by the theory of adeles.
Could somebody elaborate which features/ nuances in
the development of Nisnevich topology
reflect the striking influence on it
by the theory of adeles as stated there?
 A: He used the topology to give a cohomological interpretation of the class group/set $c(G)$ of an affine algebraic group (which is defined in terms of adeles, see below), and then to partially prove the Grothendieck-Serre conjecture. I quote Nisnevich's summary of section 2 of his thesis.
2.1. The principal role in the proofs of all results of the thesis is played by a direct relationship between the two kinds of invariants mentioned above:  the arithmetic and cohomological invariants for a general affine group scheme $G$.
The idea of such relationships already appeared in the classical isomorphism $$H^{1}(X_{Zar},\mathbb{G}_{m,X})\simeq\mathrm{Pic}(\mathbb{G}_{m,X})=c(\mathbb{G}_{m,X}),$$ which was generalized by Voskresenky and Harder to some other groups $G$. Unfortunately, in general [the] Zariski topology on $X$ is too weak to give a description of $c(G)$ and other arithmetic invariants and properties of $G$. To describe them one needs a stronger topology on $X$. For these purposes an appropriate topology is in introduced in Ch. I of the thesis. To use the topology in Ch. IV-V we define it in the following more general situation.
2.2. Let $R$ be a one dimensional regular ring, $k$ the field of fractions of $R$, $X=\mathrm{Spec}\,R$, $A$ and $A(X)$ the ring of adeles and $X$-integral adeles of $X$, $G$ an affine group scheme over $X$ with smooth general fibre $G\otimes_{R}k$, $c(G)=G(A(X))\backslash G(A)/G(k)$ the set of double cosets of $G(A)$, or in the case when $c(G)$ is a group, the class group of $G$.
2.3 Theorem-definition. There exists a Grothendieck topology $X_{cd}$ on $X$ (we shall call it the completely decomposed topology) for which one has a canonical bijection $$\alpha(G)\colon H^{1}(X_{cd},G)\simeq c(G).$$
